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Fj(is)f(the)h(o)s(ctahedron)g(and) 166 4417 y(V.)34 b(Neumann-Lara)e(pro)m(v)m(ed)j(that)f(it)e(is)h Fh(k)s Fj(-div)m(ergen)m(t)h(\(see)h([2],)f([10]\).)f(F)-8 b(or)33 b Fh(t)c Fj(=)g(5)p Fh(;)34 b(G)f Fj(is)166 4537 y(the)28 b(icosahedron,)g(and)f(M.)h(A.)g(Piza)s(~)-51 b(na)26 b(pro)m(v)m(ed)j(in)d([11])h(that)h(it)e(is)h Fh(k)s Fj(-div)m(ergen)m(t.)h(Notice)166 4658 y(that)g(these)h(graphs)f (are)g(\(1-sk)m(eletons)h(of)7 b(\))27 b(triangulations)e(of)i(the)i (sphere.)g(There)g(is)f(an)166 4778 y(in\014nite)g(n)m(um)m(b)s(er)h (of)f(lo)s(cally)f Fh(C)1327 4793 y Ft(6)1395 4778 y Fj(graphs,)i(all)e(de\014ning)h(triangulations)e(of)j(the)g(torus)g(or) 166 4898 y(the)j(Klein)f(b)s(ottle;)g(they)h(w)m(ere)i(all)29 b(pro)m(v)m(ed)34 b(to)d(b)s(e)h Fh(k)s Fj(-div)m(ergen)m(t)h(b)m(y)f (F.)g(Larri\023)-49 b(on)31 b(and)g(V.)166 5019 y(Neumann-Lara)h(in)f ([8])i(and)g([9].)166 5259 y(W)-8 b(e)41 b(will)d(describ)s(e)j(in)f (this)g(pap)s(er)h(the)g Fh(k)s Fj(-b)s(eha)m(viour)f(of)g(lo)s(cally)e Fh(C)2750 5274 y Fn(t)2820 5259 y Fj(graphs)j(for)f Fh(t)h Fg(\025)166 5380 y Fj(7.)h(F)-8 b(or)42 b(an)m(y)h(suc)m(h)g Fh(t)p Fj(,)g(there)g(exists)g(an)f(in\014nite)f(n)m(um)m(b)s(er)i(of)f (lo)s(cally)d Fh(C)2894 5395 y Fn(t)2966 5380 y Fj(graphs)k(\(see)1769 5712 y(2)p eop %%Page: 3 3 3 2 bop 166 83 a Fj(Prop)s(osition)25 b(17\).)i(Con)m(trary)g(to)g(the) g(cases)h Fh(t)g Fj(=)g(4)p Fh(;)17 b Fj(5)p Fh(;)26 b Fj(and)h(6,)g(all)e(the)i(lo)s(cally)d Fh(C)3115 98 y Fn(t)3171 83 y Fj(graphs)166 203 y(for)k Fh(t)g Fg(\025)g Fj(7)g(are)h Fh(k)s Fj(-b)s(ounded:)g(for)e(suc)m(h)j(a)f(graph)f Fh(G)g Fj(one)h(has)g Fh(k)s(G)2541 176 y Fg(\030)2542 208 y Fj(=)2647 203 y Fh(k)2701 167 y Ft(3)2740 203 y Fh(G)f Fj(\(Theorem)h(13\).)166 449 y(Our)j(pro)s(of)f(w)m(orks)j(for)d (a)h(more)f(general)h(case:)h(A)f Fi(Whitney)j(triangulation)k Fg(T)57 b Fj(of)32 b(some)166 569 y(compact)h(surface)h(is)e(one)i(suc) m(h)g(that)f(an)m(y)h(triangle)d(of)i(its)f(underlying)h(graph)g Fh(G)c Fj(=)f Fg(T)3414 584 y Ft(1)166 689 y Fj(is)j(a)g(face)h(of)f Fg(T)26 b Fj(.)31 b(Suc)m(h)i(a)e(triangulation)e Fg(T)57 b Fj(is)31 b(fully)f(determined)h(b)m(y)i(its)e(1-sk)m(eleton)h Fh(G)p Fj(,)166 810 y(and)37 b(these)h(graphs)f(admit)e(a)i(nice)g(c)m (haracterization)e(in)i(com)m(binatorial)c(terms:)k(eac)m(h)166 930 y(op)s(en)i(neigh)m(b)s(ourho)s(o)s(d)e(in)g(suc)m(h)j(a)e(graph)g Fh(G)h Fj(is)e(either)i(a)f(cycle)g(or)g(a)h(path)f(and)g Fg(T)64 b Fj(is)166 1050 y(closed)36 b(if)e(and)i(only)f(if)g Fh(G)h Fj(is)f Fi(lo)-5 b(c)g(al)5 b(ly)37 b(cyclic,)e Fj(i.e.)g(eac)m(h)i(op)s(en)f(neigh)m(b)s(ourho)s(o)s(d)f(in)g Fh(G)g Fj(is)166 1171 y(a)d(cycle)h(\(see)g Fg(x)p Fj(4.1\).)f(Notice)g (that)g(a)f(lo)s(cally)f Fh(C)1925 1186 y Fn(t)1986 1171 y Fj(graph)i(is)g(just)h(a)f(lo)s(cally)d(cyclic)j(graph)166 1291 y(whic)m(h)40 b(is)f(regular)g(of)g(degree)h Fh(t)p Fj(,)g(and)f(th)m(us)i(Theorem)f(13)f(will)e(b)s(e)i(a)h(consequence)i (of)166 1412 y(Theorem)29 b(12:)f Fh(k)s(G)885 1384 y Fg(\030)886 1416 y Fj(=)990 1412 y Fh(k)1044 1375 y Ft(3)1084 1412 y Fh(G)g Fj(for)g(ev)m(ery)i(lo)s(cally)c(cyclic)i(graph)g Fh(G)g Fj(with)g(minim)m(um)d(degree)166 1532 y Fh(\016)t Fj(\()p Fh(G)p Fj(\))i Fg(\025)i Fj(7.)166 1777 y(P)m(art)35 b(of)f(the)h(pro)s(of)e(w)m(orks)j(in)e(ev)m(en)i(greater)f(generalit)m (y)-8 b(.)33 b(The)j Fi(lo)-5 b(c)g(al)36 b(girth)42 b Fj(of)34 b(a)g(graph)166 1897 y Fh(G)42 b Fj(is)f(the)h(minim)m(um)d (of)i(the)i(girths)e(of)g(the)h(op)s(en)g(neigh)m(b)s(ourho)s(o)s(ds)g (in)f Fh(G)p Fj(.)h(W)-8 b(e)42 b(will)166 2018 y(sho)m(w)35 b(in)e Fg(x)p Fj(3)g(that)h Fh(G)f Fj(is)h(ev)m(en)m(tually)g Fh(k)s Fj(-p)s(erio)s(dic)d(whenev)m(er)37 b(its)c(lo)s(cal)e(girth)i (is)g(at)g(least)166 2138 y(7)c(\(Theorem)g(8\).)f(In)h(particular)e (this)i(implies)d(the)j Fh(k)s Fj(-b)s(oundedness)i(of)d(the)h (underlying)166 2259 y(graph)42 b(of)g(an)m(y)h(Whitney)g (triangulation)c(of)j(a)h(compact)f(surface)h(\(with)f(or)g(without)166 2379 y(b)s(order\))35 b(pro)m(vided)g(that)g(the)h(minim)m(um)31 b(degree)36 b(of)f(the)g(in)m(terior)f(v)m(ertices)i(is)e(at)h(least) 166 2499 y(7)i(\(Theorem)g(16\).)g(F)-8 b(or)36 b(instance,)i(if)e Fh(G)h Fj(underlies)f(a)h(Whitney)h(triangulation)c(of)i(the)166 2620 y(disk,)c(then)g Fh(G)f Fj(is)g Fh(k)s Fj(-b)s(ounded)h(if)e(the)i (minim)m(um)c(degree)33 b(of)e(the)g(in)m(terior)f(v)m(ertices)j(is)e (at)166 2740 y(least)h(7,)h(but)f(w)m(e)i(conjecture)g(a)e(m)m(uc)m(h)h (stronger)g(result:)166 2985 y Fk(Conjecture)38 b(1)48 b Fi(L)-5 b(et)41 b Fh(G)f Fi(b)-5 b(e)39 b(the)h(underlying)g(gr)-5 b(aph)39 b(of)g(a)h(Whitney)g(triangulation)g(of)166 3106 y(the)35 b(disk.)f(Then)g Fh(G)h Fi(is)g Fh(k)s Fi(-nul)5 b(l.)166 3351 y Fj(Notice)34 b(that)h(here)h(no)e (restrictions)h(on)f(the)i(minim)m(um)31 b(degree)36 b(are)f(made.)f(A)h(related)166 3471 y(conjecture)f(is:)166 3716 y Fk(Conjecture)k(2)48 b Fi(L)-5 b(et)26 b Fh(G)h Fg(6)p Fj(=)h Fh(K)1295 3731 y Ft(4)1359 3716 y Fi(b)-5 b(e)25 b(the)g(underlying)g(gr)-5 b(aph)24 b(of)h(a)g(Whitney)g (triangulation)166 3837 y(of)35 b(the)g(spher)-5 b(e.)34 b(Then)g Fh(G)g Fi(is)h Fh(k)s Fi(-diver)-5 b(gent.)166 4408 y Fk(2)112 b(Clique-Helly)35 b(graphs)166 4774 y Fj(A)k(family)e Fg(C)45 b Fj(of)39 b(sets)i(is)d(said)h(to)g(satisfy)g (the)h Fi(Hel)5 b(ly)41 b(pr)-5 b(op)g(erty)48 b Fj(if)38 b(the)i(in)m(tersection)f(of)166 4894 y(an)m(y)k(subfamily)e Fg(F)52 b Fj(of)43 b(m)m(utually)e(in)m(tersecting)h(mem)m(b)s(ers)h (of)f Fg(C)49 b Fj(is)42 b(non-empt)m(y)-8 b(.)43 b(The)166 5014 y(follo)m(wing)30 b(lemma)g(pro)m(vides)k(an)e(in)m(teresting)g (example:)166 5259 y Fk(Lemma)37 b(3)49 b Fi(L)-5 b(et)29 b Fh(G)f Fi(b)-5 b(e)29 b(a)f(gr)-5 b(aph)28 b(and)g(let)h Fg(C)35 b Fi(b)-5 b(e)28 b(any)h(family)f(of)g(e)-5 b(dges)28 b(of)g Fh(G)h Fi(such)f(that)h(no)166 5380 y(e)-5 b(dge)34 b(of)h Fg(C)41 b Fi(is)35 b(c)-5 b(ontaine)g(d)33 b(in)i(a)g(triangle)f (of)h Fh(G)p Fi(.)f(Then)g Fg(C)42 b Fi(satis\014es)34 b(the)h(Hel)5 b(ly)35 b(pr)-5 b(op)g(erty.)1769 5712 y Fj(3)p eop %%Page: 4 4 4 3 bop 166 83 a Fk(PR)m(OOF.)49 b Fj(Fix)40 b Fg(F)50 b(\022)42 b(C)47 b Fj(suc)m(h)42 b(that)f(an)m(y)g(t)m(w)m(o)g(edges)h (in)e Fg(F)50 b Fj(meet.)40 b(Assume)i(that)e Fg(F)166 203 y Fj(con)m(tains)30 b(at)g(least)g(t)m(w)m(o)h(edges)h Fh(e)1375 218 y Ft(1)1442 203 y Fj(=)27 b Fg(f)p Fh(x;)17 b(a)p Fg(g)31 b Fj(and)f Fh(e)2058 218 y Ft(2)2125 203 y Fj(=)e Fg(f)p Fh(x;)17 b(b)p Fg(g)p Fj(.)31 b(An)m(y)g(other)f(edge)h Fh(e)d Fg(2)g(F)166 324 y Fj(con)m(tains)40 b(the)g(v)m(ertex)h Fh(x)p Fj(,)f(since)g(otherwise)g Fh(e)g Fj(=)f Fg(f)p Fh(a;)17 b(b)p Fg(g)40 b Fj(w)m(ould)f(b)s(e)h(con)m(tained)f(in)g(the) 166 444 y(triangle)31 b Fg(f)p Fh(x;)17 b(a;)g(b)p Fg(g)p Fj(.)98 b Ff(2)166 860 y Fj(A)40 b(graph)f Fh(H)47 b Fj(is)39 b(said)g(to)g(b)s(e)h Fi(clique-Hel)5 b(ly)48 b Fj(if)38 b(the)i(family)d Fg(C)46 b Fj(of)39 b(all)f(the)i(cliques)f (of)g Fh(H)166 981 y Fj(satis\014es)k(the)g(Helly)e(prop)s(ert)m(y)-8 b(.)43 b(Giv)m(en)f(an)m(y)h(graph)f Fh(H)50 b Fj(the)43 b(relation)d(of)i Fi(domination)166 1101 y Fj(among)29 b(the)h(v)m(ertices)i(of)d Fh(H)38 b Fj(is)30 b(de\014ned)h(as)g(follo) m(ws:)d(the)j(v)m(ertex)h Fh(v)i Fj(is)29 b(said)h(to)g Fi(dominate)166 1221 y Fj(the)41 b(v)m(ertex)h Fh(u)d Fj(if)g Fh(N)10 b Fj([)p Fh(u)p Fj(])41 b Fg(\022)h Fh(N)10 b Fj([)p Fh(v)t Fj(],)40 b(where)i Fh(N)10 b Fj([)p Fh(v)t Fj(])41 b(=)g Fh(N)10 b Fj(\()p Fh(v)t Fj(\))27 b Fg([)h(f)p Fh(v)t Fg(g)39 b Fj(is)h(the)h Fi(close)-5 b(d)41 b(neigh-)166 1342 y(b)-5 b(ourho)g(o)g(d)51 b Fj(of)40 b Fh(v)t Fj(.)h(Th)m(us,)h Fh(v)j Fj(dominates)40 b Fh(u)g Fj(if)g(and)h(only)g(if)e(either)i Fh(v)46 b Fj(=)c Fh(u)e Fj(or)h Fh(v)j Fj(and)d Fh(u)166 1462 y Fj(are)g(neigh)m(b)s(ours)f(and,)h(apart)f(from)g Fh(v)t Fj(,)g(ev)m(ery)i(neigh)m(b)s(our)f(of)f Fh(u)g Fj(is)g(also)g(a)g(neigh)m(b)s(our)166 1583 y(of)e Fh(v)t Fj(.)h(The)h(relation)d(of)h(domination)e(is)j(a)f(preorder)i(in)e Fh(V)21 b Fj(\()p Fh(H)8 b Fj(\))38 b(\(i.e.,)h(this)f(relation)f(is) 166 1703 y(re\015exiv)m(e)j(and)f(transitiv)m(e\).)f(As)h(usual)f(with) g(preorders,)i(the)f Fi(mutual)i(domination)j Fj(is)166 1823 y(an)25 b(equiv)-5 b(alence)26 b(relation)e(in)h Fh(V)c Fj(\()p Fh(H)8 b Fj(\))p Fh(;)25 b Fj(and)h(the)g(domination)d (relation)g(induces)k(a)e(partial)166 1944 y(order)30 b(on)g(the)h(quotien)m(t)f(set)h(of)f(all)e(the)i(classes)h(of)f(m)m (utual)f(domination)e(of)j(v)m(ertices)h(of)166 2064 y Fh(H)8 b Fj(.)37 b(In)g(this)g(language,)f(the)h(results)h(\(Satz)f (1)g(and)g(Satz)g(2\))g(of)f(Escalan)m(te's)j(pap)s(er)e([2])166 2184 y(whic)m(h)c(w)m(e)h(will)c(use)k(are:)166 2414 y Fk(Theorem)j(4)49 b Fi(If)29 b Fh(H)38 b Fi(is)29 b(clique-Hel)5 b(ly)30 b(then)f(its)h(clique)g(gr)-5 b(aph)29 b Fh(k)s(H)37 b Fi(is)30 b(also)f(clique-Hel)5 b(ly.)166 2643 y Fk(Theorem)37 b(5)49 b Fi(If)34 b Fh(H)42 b Fi(is)34 b(clique-Hel)5 b(ly)34 b(then)h(its)f(se)-5 b(c)g(ond)34 b(clique)g(gr)-5 b(aph)34 b Fh(k)2862 2607 y Ft(2)2901 2643 y Fh(H)42 b Fi(is)35 b(isomor-)166 2763 y(phic)j(to)h(the)f(sub)-5 b(gr)g(aph)p 1067 2685 79 4 v 39 w Fh(E)6 b Fj(\()p Fh(H)i Fj(\))38 b Fi(of)g Fh(H)46 b Fi(induc)-5 b(e)g(d)38 b(by)h(any)f (system)h(of)f(r)-5 b(epr)g(esentatives)38 b(of)166 2884 y(the)d(maximal)e(classes)h(of)h(mutual)g(domination.)166 3113 y Fj(The)45 b(second)g(of)f(the)g(ab)s(o)m(v)m(e)h(results)f (generalize)g(the)g(oldest)g(result)g(ab)s(out)f(iterated)166 3233 y(clique)32 b(graphs,)h(whic)m(h)g(is)f(due)i(to)e(S.)h(T.)g (Hedetniemi)e(and)i(P)-8 b(.)33 b(J.)f(Slater)g([5]:)166 3462 y Fk(Prop)s(osition)k(6)48 b Fi(If)42 b Fh(H)49 b Fi(is)41 b(c)-5 b(onne)g(cte)g(d,)41 b(triangleless,)g(and)g(with)g (at)h(le)-5 b(ast)42 b Fj(3)f Fi(vertic)-5 b(es,)166 3583 y(then)35 b Fh(k)437 3547 y Ft(2)476 3583 y Fh(H)592 3555 y Fg(\030)593 3587 y Fj(=)698 3583 y Fh(H)29 b Fg(\000)23 b(f)p Fh(v)31 b Fg(j)c Fj(deg)r(\()p Fh(v)t Fj(\))g(=)h(1)p Fg(g)p Fi(.)166 3999 y Fk(PR)m(OOF.)49 b Fj(By)31 b(Lemma)f(3,)h Fh(H)39 b Fj(is)30 b(clique-Helly)-8 b(.)30 b(The)i(v)m(ertices)g(whic) m(h)g(are)f(dominated)166 4119 y(b)m(y)37 b(another)g(are)g(those)g(of) f(degree)h(1.)g(The)g(relation)e(of)h(m)m(utual)f(domination)f(is)i (triv-)166 4240 y(ial.)96 b Ff(2)166 4656 y Fj(The)30 b(follo)m(wing)c(result)j(is)g(an)g(easy)h(consequence)i(of)c(Theorems) i(4)f(and)g(5.)f(It)h(is)g(all)e(that)166 4776 y(w)m(e)34 b(will)c(need)k(of)e(Escalan)m(te's)h(pap)s(er)g([2]:)166 5005 y Fk(Prop)s(osition)j(7)48 b Fi(L)-5 b(et)36 b Fh(H)42 b Fi(b)-5 b(e)35 b(a)f(clique-Hel)5 b(ly)35 b(gr)-5 b(aph.)34 b(Then:)193 5235 y(\(1\))49 b Fh(H)42 b Fi(is)35 b(eventual)5 b(ly)35 b Fh(k)s Fi(-p)-5 b(erio)g(dic)34 b(of)g(p)-5 b(erio)g(d)34 b Fj(1)h Fi(or)g Fj(2)p Fi(,)f(and)193 5355 y(\(2\))49 b(If)34 b(e)-5 b(ach)34 b(vertex)h(of)g Fh(H)42 b Fi(only)35 b(dominates)f(itself,)g(then)h Fh(k)2434 5319 y Ft(2)2473 5355 y Fh(H)2589 5327 y Fg(\030)2590 5359 y Fj(=)2695 5355 y Fh(H)8 b Fi(.)99 b Ff(2)1769 5712 y Fj(4)p eop %%Page: 5 5 5 4 bop 166 83 a Fk(3)112 b(Graphs)39 b(with)d(lo)s(cal)g(girth)g(at)i (least)e Fj(7)166 426 y(Recall)42 b(that)h(the)h Fi(girth)50 b Fh(g)t Fj(\()p Fh(G)p Fj(\))43 b(of)f(a)i(graph)f Fh(G)g Fj(is)g(the)g(minim)m(um)d(length)j(of)g(a)g(cycle)166 546 y(in)h Fh(G)g Fj(\(if)f Fh(G)i Fj(do)s(es)g(not)f(ha)m(v)m(e)i (cycles,)f(then)g Fh(g)t Fj(\()p Fh(G)p Fj(\))i(=)h Fg(1)p Fj(\).)c(If)h Fh(v)51 b Fg(2)e Fh(V)21 b Fj(\()p Fh(G)p Fj(\),)45 b(the)f Fi(lo)-5 b(c)g(al)166 666 y(girth)35 b(of)52 b Fh(G)32 b Fi(at)42 b Fh(v)36 b Fj(is)c(the)g(girth)f(of)h (the)h(op)s(en)f(neigh)m(b)s(ourho)s(o)s(d)f(of)h Fh(v)k Fj(in)31 b Fh(G)p Fj(;)h(in)g(sym)m(b)s(ols,)166 787 y(lg)243 810 y Fn(v)284 787 y Fj(\()p Fh(G)p Fj(\))k(=)f Fh(g)t Fj(\()p Fh(N)10 b Fj(\()p Fh(v)t Fj(\)\).)36 b(The)j Fi(lo)-5 b(c)g(al)38 b(girth)47 b Fj(of)36 b(the)i(graph)f Fh(G)g Fj(is)g(the)g(minim)m(um)d(of)j(all)e(the)166 907 y(lo)s(cal)c(girths)h(of)g Fh(G)p Fj(,)h(i.e.,)g(lg\()p Fh(G)p Fj(\))28 b(=)g(min)o Fg(f)p Fj(lg)1759 931 y Fn(v)1800 907 y Fj(\()p Fh(G)p Fj(\))g Fg(j)g Fh(v)k Fg(2)c Fh(V)22 b Fj(\()p Fh(G)p Fj(\))p Fg(g)p Fj(.)32 b(F)-8 b(or)32 b(instance)i(an)m(y)f(tree)166 1028 y(has)27 b(in\014nite)e(lo)s(cal)f (girth,)i(but)g(also)g(a)g(cycle)h(\(or)f(a)g(cycle)h(with)f(a)g (diagonal\))e(has)j(in\014nite)166 1148 y(lo)s(cal)j(girth.)i(The)h (main)e(purp)s(ose)j(of)e(this)g(section)h(is)f(to)g(pro)m(v)m(e)i(the) f(follo)m(wing)d(result:)166 1370 y Fk(Theorem)37 b(8)49 b Fi(If)37 b(the)h(lo)-5 b(c)g(al)37 b(girth)g(of)g(the)h(gr)-5 b(aph)37 b Fh(G)h Fi(satis\014es)f Fj(lg\()p Fh(G)p Fj(\))32 b Fg(\025)i Fj(7)p Fi(,)j(then)g Fh(k)s(G)h Fi(is)166 1490 y(clique-Hel)5 b(ly.)35 b(In)f(p)-5 b(articular,)35 b Fh(G)f Fi(is)h(eventual)5 b(ly)35 b Fh(k)s Fi(-p)-5 b(erio)g(dic)34 b(of)g(p)-5 b(erio)g(d)34 b Fj(1)h Fi(or)g Fj(2)p Fi(.)166 1713 y Fj(That)k Fh(G)g Fj(is)f(ev)m(en)m(tually)i Fh(k)s Fj(-p)s(erio)s(dic)c(of)j(p)s(erio)s(d)e(1)i(or)g(2)f(is)g (immediate)f(from)g(7\(1\),)h(so)166 1833 y(w)m(e)31 b(need)f(to)f(pro)m(v)m(e)i(only)e(that)g Fh(k)s(G)g Fj(is)g(clique-Helly)-8 b(.)28 b(In)h(order)h(to)f(do)g(this,)h(w)m(e)g (will)d(\014rst)166 1953 y(study)37 b(the)e(cliques)g(of)g Fh(k)s(G)g Fj(in)g Fg(x)p Fj(3.1,)g(and)h(then)f(in)g Fg(x)p Fj(3.2)g(w)m(e)h(will)d(reduce)k(the)f(pro)s(of)e(to)166 2074 y(a)e(simple)e(statemen)m(t)j(ab)s(out)f(families)d(of)j (triangles)e(and)i(v)m(ertices)i(in)d(a)h(graph)g Fh(G)g Fj(with)166 2194 y(lg\()p Fh(G)p Fj(\))c Fg(\025)g Fj(7.)166 2546 y Fi(3.1)99 b(The)34 b(cliques)h(of)f Fh(k)s(G)166 2889 y Fj(Assuming)48 b(only)h(that)f(lg\()p Fh(G)p Fj(\))56 b Fg(\025)f Fj(4)49 b(\(i.e.,)f(that)h Fh(G)g Fj(con)m(tains)f(no)h (tetrahedron)g Fh(K)3376 2904 y Ft(4)3416 2889 y Fj(\))166 3009 y(the)39 b(largest)e(p)s(ossible)h(order)h(for)e(a)i(clique)e(in)h Fh(G)g Fj(is)g(3,)g(and)h(so)f(the)h(cliques)g(of)e Fh(G)i Fj(are)166 3130 y(its)h(triangles)f(and)i(those)g(of)f(its)g(edges)i (that)e(are)g(not)h(con)m(tained)f(in)g(some)g(triangle.)166 3250 y(The)47 b(clique)e(graph)h Fh(k)s(G)g Fj(has)g(these)h(cliques)f (as)g(v)m(ertices)h(and)f(the)g(adjacencies)h(are)166 3370 y(non-empt)m(y)42 b(in)m(tersections.)f(Note)h(that)f(an)g(edge)h (and)g(a)f(triangle)e(\(or)i(an)h(edge)g(and)166 3491 y(a)35 b(di\013eren)m(t)h(edge\))f(can)h(ha)m(v)m(e)h(only)d(one)i(v)m (ertex)h(in)e(common,)f(whereas)i(t)m(w)m(o)g(di\013eren)m(t)166 3611 y(triangles)h(can)i(meet)f(either)g(at)g(a)g(v)m(ertex)i(or)e(at)g (an)h(edge.)g(It)f(will)e(b)s(e)j(con)m(v)m(enien)m(t)h(to)166 3732 y(distinguish)31 b(t)m(w)m(o)j(kinds)f(of)f(cliques)g(of)g Fh(k)s(G)p Fj(:)h(the)g Fi(stars)41 b Fj(and)32 b(the)h Fi(ne)-5 b(ckties)p Fj(.)166 3954 y(Giv)m(en)38 b(a)g(v)m(ertex)i Fh(v)i Fj(of)c(an)g(arbitrary)f(graph)h Fh(G)p Fj(,)h(the)f Fi(star)49 b Fj(of)38 b Fh(v)k Fj(is)c(the)g(set)h(of)f(all)e(the)166 4074 y(cliques)29 b(of)g Fh(G)h Fj(whic)m(h)g(con)m(tain)f Fh(v)t Fj(,)g(i.e.)g Fh(v)1624 4038 y Fe(\003)1691 4074 y Fj(=)e Fg(f)p Fh(Q)h Fg(2)g Fh(V)22 b Fj(\()p Fh(k)s(G)p Fj(\))27 b Fg(j)h Fh(v)j Fg(2)d Fh(Q)p Fg(g)p Fj(.)i(Then)g Fh(v)3070 4038 y Fe(\003)3139 4074 y Fj(induces)166 4195 y(a)i(complete)f(subgraph)i(of)f Fh(k)s(G)g Fj(whic)m(h)h(ma)m(y)e(or)h (ma)m(y)g(not)g(b)s(e)g(a)g(clique)g(of)g Fh(k)s(G)p Fj(.)g(W)-8 b(e)32 b(call)166 4315 y Fh(v)44 b Fj(a)c Fi(c)-5 b(enter)50 b Fj(of)40 b Fh(v)824 4279 y Fe(\003)903 4315 y Fj(\(the)h(cen)m(ters)h(of)d Fh(v)1622 4279 y Fe(\003)1702 4315 y Fj(are)h(all)e(t)m(wins\).)i(Notice)g(that)g(a)g (clique)g Fg(Q)g Fj(of)166 4435 y Fh(k)s(G)i Fj(is)g(a)f(star)i(if)e (and)h(only)f(if)g(the)i(in)m(tersection)f(of)f(all)f(the)j(elemen)m (ts)f(of)g Fg(Q)g Fj(\(whic)m(h)166 4556 y(are)d(cliques)f(of)h Fh(G)p Fj(\))f(is)g(non-empt)m(y:)h(there)h(is)e(a)g(v)m(ertex)j(of)d Fh(G)g Fj(whic)m(h)h(is)g(con)m(tained)f(in)166 4676 y(all)h(the)j(elemen)m(ts)g(of)f Fg(Q)p Fj(.)h(The)g(set)g(of)f(all)f (the)i(stars)g(whic)m(h)g(are)f(cliques)h(of)f Fh(k)s(G)g Fj(is)g(a)166 4796 y(set)i(of)e(v)m(ertices)j(in)d Fh(k)988 4760 y Ft(2)1028 4796 y Fh(G)p Fj(,)h(and)g(t)m(w)m(o)h(di\013eren)m(t) f(stars)h Fh(u)2261 4760 y Fe(\003)2342 4796 y Fj(and)f Fh(v)2592 4760 y Fe(\003)2673 4796 y Fj(ha)m(v)m(e)i(a)d(non-empt)m(y) 166 4917 y(in)m(tersection)31 b(\(i.e.)g(are)g(adjacen)m(t)h(in)f Fh(k)1599 4881 y Ft(2)1638 4917 y Fh(G)p Fj(\))g(if)f(and)i(only)f(if)f Fh(u)g Fj(and)i Fh(v)j Fj(are)c(neigh)m(b)s(ours)g(in)166 5037 y Fh(G)p Fj(.)i(The)g(Helly)f(prop)s(ert)m(y)h(holds)f(for)g (stars:)166 5259 y Fk(Lemma)37 b(9)49 b Fi(L)-5 b(et)34 b Fh(G)f Fi(b)-5 b(e)34 b(any)f(gr)-5 b(aph)33 b(and)g(let)h Fg(F)43 b Fi(b)-5 b(e)33 b(a)h(family)f(of)g(mutual)5 b(ly)34 b(interse)-5 b(cting)166 5380 y(stars)37 b(of)g(vertic)-5 b(es)36 b(of)h Fh(G)p Fi(.)f(Then)h(the)g(interse)-5 b(ction)36 b(of)g(al)5 b(l)37 b(the)g(memb)-5 b(ers)36 b(of)g Fg(F)47 b Fi(is)36 b(non-)1769 5712 y Fj(5)p eop %%Page: 6 6 6 5 bop 166 83 a Fi(empty.)166 431 y Fk(PR)m(OOF.)49 b Fj(If)35 b Fg(Q)f(2)f(F)10 b Fj(,)35 b(then)i(there)f(is)f(some)h(v)m (ertex)h Fh(v)g Fg(2)d Fh(V)21 b Fj(\()p Fh(G)p Fj(\))36 b(suc)m(h)h(that)e Fg(Q)f Fj(=)e Fh(v)3387 394 y Fe(\003)3427 431 y Fj(.)166 551 y(Since)i(an)m(y)g(t)m(w)m(o)h(mem)m(b)s(ers)e(of)h Fg(F)43 b Fj(ha)m(v)m(e)35 b(a)e(non-empt)m(y)h(in)m(tersection,)g(the) g(subgraph)h Fh(C)166 671 y Fj(of)e Fh(G)g Fj(induced)g(b)m(y)h(the)g (cen)m(ters)h(of)d(the)i(stars)f(in)g Fg(F)42 b Fj(is)33 b(complete.)f(An)m(y)i(clique)f Fh(Q)g Fj(of)g Fh(G)166 792 y Fj(whic)m(h)g(con)m(tains)g Fh(C)39 b Fj(is)32 b(then)i(a)e(mem)m(b)s(er)g(of)g(all)e(the)j(stars)g(in)f Fg(F)10 b Fj(.)97 b Ff(2)166 1139 y Fj(An)m(y)42 b(clique)f(of)f Fh(k)s(G)h Fj(whic)m(h)h(is)e(not)h(a)g(star)g(will)d(b)s(e)k(said)e (to)h(b)s(e)g(a)g Fi(ne)-5 b(cktie)p Fj(.)40 b(A)h(clique)166 1260 y Fg(Q)f Fj(of)g Fh(k)s(G)g Fj(is)g(a)f(nec)m(ktie)j(i\013)d Fg(\\Q)i Fj(=)f Fd(?)p Fj(,)h(i\013)e(for)g(ev)m(ery)j(v)m(ertex)g (whic)m(h)f(is)e(con)m(tained)h(in)166 1380 y(some)33 b(elemen)m(t)h(of)f Fg(Q)h Fj(there)g(exists)g(another)g(elemen)m(t)f (of)g Fg(Q)h Fj(whic)m(h)g(do)s(es)g(not)f(con)m(tain)166 1500 y(that)g(v)m(ertex.)i(W)-8 b(e)33 b(will)e(giv)m(e)i(no)m(w)g(the) h(construction)f(of)f(certain)h(nec)m(kties.)h(These)h(will)166 1621 y(indeed)c(b)s(e)g(nec)m(kties)i(if)c(lg\()p Fh(G)p Fj(\))f Fg(\025)g Fj(4,)j(and)g(they)g(will)e(b)s(e)i Fi(al)5 b(l)41 b Fj(the)31 b(nec)m(kties)h(if)e(lg\()p Fh(G)p Fj(\))e Fg(\025)g Fj(5.)166 1842 y(Consider)33 b(an)g Fi(inner)43 b Fj(triangle)31 b Fh(T)42 b Fj(=)28 b Fg(f)p Fh(a;)17 b(b;)g(c)p Fg(g)33 b Fj(of)f Fh(G)p Fj(,)h(i.e.,)g(for)f(eac)m(h)i(of)f(the)g(three)h(edges)166 1962 y(of)e Fh(T)47 b Fj(there)33 b(exists)h(at)e(least)g(one)h (triangle)e Fh(T)1855 1926 y Fe(0)1911 1962 y Fj(of)h Fh(G)h Fj(suc)m(h)h(that)f Fh(T)2635 1926 y Fe(0)2686 1962 y Fg(6)p Fj(=)27 b Fh(T)47 b Fj(and)33 b Fh(T)3154 1926 y Fe(0)3209 1962 y Fj(meets)166 2082 y Fh(T)41 b Fj(in)26 b(that)h(edge.)h(The)g Fi(ne)-5 b(cktie)29 b(of)78 b Fh(G)27 b Fi(c)-5 b(enter)g(e)g(d)29 b(at)37 b Fh(T)k Fj(is)26 b(the)i(collection)d Fg(Q)2953 2097 y Fn(T)3035 2082 y Fj(consisting)166 2203 y(of)33 b Fh(T)47 b Fj(and)34 b(all)d(the)j(triangles)e(that)h(meet)h Fh(T)47 b Fj(at)33 b(some)g(edge.)i(The)f(triangle)e Fh(T)47 b Fj(is)33 b(called)166 2323 y(the)g Fi(c)-5 b(enter)43 b Fj(of)32 b Fg(Q)827 2338 y Fn(T)915 2323 y Fj(and)h(the)g(remaining)d(triangles) h(of)h Fg(Q)2316 2338 y Fn(T)2404 2323 y Fj(are)h(its)f Fi(e)-5 b(ars)p Fj(.)166 2544 y(Let)26 b(us)f(sho)m(w)i(that)e Fg(Q)971 2559 y Fn(T)1052 2544 y Fj(is)f(indeed)i(a)f(nec)m(ktie)h(if)f (lg\()p Fh(G)p Fj(\))i Fg(\025)h Fj(4:)e(Recall)d(that)i(ev)m(ery)j (triangle)166 2664 y(of)38 b Fh(G)g Fj(is)g(a)g(clique)f(of)h Fh(G)p Fj(,)g(i.e.)g(a)g(v)m(ertex)i(of)e Fh(k)s(G)p Fj(.)g(Since)g(an)m(y)h(triangle)e(in)g Fg(Q)3010 2679 y Fn(T)3104 2664 y Fj(con)m(tains)166 2785 y(at)32 b(least)g(an)h(edge) g(of)f Fh(T)46 b Fj(and)32 b(an)m(y)h(t)m(w)m(o)h(edges)f(of)f Fh(T)46 b Fj(meet)32 b(in)g(at)g(least)g(one)h(v)m(ertex,)h Fg(Q)3398 2800 y Fn(T)166 2905 y Fj(induces)k(a)f(complete)g(subgraph)h (of)f Fh(k)s(G)p Fj(.)h(Let)f(us)h(sho)m(w)h(that)e Fg(Q)2614 2920 y Fn(T)2707 2905 y Fj(is)g(a)g(clique)g(of)g Fh(k)s(G)p Fj(,)166 3026 y(assuming)30 b(to)h(the)g(con)m(trary)h(that)e(there)i (is)e(some)h(clique)f Fh(Q)h Fj(of)g Fh(G)g Fj(whic)m(h)g(is)f(not)h (in)f Fg(Q)3398 3041 y Fn(T)166 3146 y Fj(but)35 b(meets)h(ev)m(ery)h (triangle)c(in)i Fg(Q)1445 3161 y Fn(T)1500 3146 y Fj(.)g(Note)g(that)g Fh(Q)h Fj(can)f(not)g(meet)g Fh(T)49 b Fj(in)34 b(an)h(edge,)h(for)166 3266 y(then)d Fh(Q)f Fj(w)m(ould)g(b)s(e)g(a)f(triangle)f(and)j(so)f(a) f(mem)m(b)s(er)h(of)f Fg(Q)2304 3281 y Fn(T)2360 3266 y Fj(.)g(Th)m(us,)j Fh(Q)e Fj(meets)h Fh(T)45 b Fj(in)31 b(just)166 3387 y(one)c(v)m(ertex,)i(sa)m(y)f Fh(c)p Fj(.)f(Let)g Fh(T)1155 3351 y Fe(0)1206 3387 y Fj(=)g Fg(f)p Fh(a;)17 b(b;)g(c)1581 3351 y Fe(0)1604 3387 y Fg(g)27 b Fj(b)s(e)g(a)g(triangle)e(of)h Fh(G)h Fj(whic)m(h)h(meets)f Fh(T)40 b Fj(precisely)166 3507 y(in)35 b(the)i(edge)f Fg(f)p Fh(a;)17 b(b)p Fg(g)p Fj(.)36 b(Hence,)i Fh(T)1371 3471 y Fe(0)1427 3507 y Fg(2)c(Q)1608 3522 y Fn(T)1664 3507 y Fj(.)i(Since)g Fh(Q)g Fj(has)h(to)e(meet)h Fh(T)2711 3471 y Fe(0)2734 3507 y Fj(,)g(and)g Fh(Q)h Fj(con)m(tains)166 3627 y(neither)43 b Fh(a)g Fj(nor)f Fh(b)p Fj(,)i(then)f Fh(Q)g Fj(m)m(ust)g(con)m(tain)g Fh(c)1897 3591 y Fe(0)1920 3627 y Fj(.)g(But)g(then)g Fg(f)p Fh(b;)17 b(c;)g(c)2689 3591 y Fe(0)2712 3627 y Fg(g)43 b Fj(is)f(a)h(triangle)d(in)166 3748 y Fh(N)10 b Fj(\()p Fh(a)p Fj(\),)42 b(con)m(tradicting)e(that)g (lg)q(\()p Fh(G)p Fj(\))h Fg(\025)i Fj(4.)e(This)g(con)m(tradiction)f (sho)m(ws)i(that)f Fg(Q)3202 3763 y Fn(T)3298 3748 y Fj(is)g(a)166 3868 y(clique)33 b(of)g Fh(k)s(G)h Fj(for)e(eac)m(h)j (inner)e(triangle)f Fh(T)47 b Fj(of)33 b Fh(G)p Fj(.)g(Since)h(the)g 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Fk(PR)m(OOF.)49 b Fj(Let)34 b Fg(Q)h Fj(b)s(e)f(an)m(y)h(clique)f(of) g Fh(k)s(G)g Fj(whic)m(h)h(is)e(not)i(a)f(star.)g(W)-8 b(e)35 b(will)c(sho)m(w)36 b(that)166 5380 y Fg(Q)c Fj(=)g Fg(Q)468 5395 y Fn(T)559 5380 y Fj(for)i(some)h(inner)g(triangle)e Fh(T)49 b Fj(of)35 b Fh(G)p Fj(.)g(The)h(elemen)m(ts)f(of)g Fg(Q)p Fj(,)h(b)s(eing)e(cliques)h(of)1769 5712 y(6)p eop %%Page: 7 7 7 6 bop 166 83 a Fh(G)p Fj(,)28 b(are)g(either)g(triangles)f(of)h Fh(G)g Fj(or)g(edges)h(of)e Fh(G)i Fj(whic)m(h)f(are)g(not)g(con)m (tained)h(in)e(triangles.)166 203 y(It)35 b(can)h(not)f(b)s(e)h(the)f (case)h(that)g(all)d(the)i(elemen)m(ts)h(of)f Fg(Q)g Fj(are)h(edges,)g(since)g(b)m(y)g(Lemma)166 324 y(3)c Fg(Q)h Fj(w)m(ould)g(b)s(e)g(a)f(star)g(in)g(that)h(case.)g(Hence)h (there)f(is)f(a)h(triangle)d Fh(T)42 b Fj(=)27 b Fg(f)p Fh(a;)17 b(b;)g(c)p Fg(g)28 b(2)g(Q)p Fj(.)166 552 y(No)m(w)42 b(w)m(e)h(claim)c(that)i(eac)m(h)h(elemen)m(t)g(of)f Fg(Q)g Fj(is)g(a)h(triangle.)d(Indeed,)k(if)e Fg(Q)h Fj(con)m(tained)166 673 y(some)c(edge)h Fh(E)6 b Fj(,)39 b(then)g(\(sa)m(y\))g Fh(E)k Fj(=)38 b Fg(f)p Fh(a;)17 b(d)p Fg(g)37 b Fj(for)h(some)g(v)m(ertex)i Fh(d)e Fj(not)g(in)g Fh(T)14 b Fj(.)38 b(Since)h Fg(Q)f Fj(is)166 793 y(a)e(nec)m(ktie,)i (there)f(m)m(ust)g(b)s(e)g(a)f Fh(Q)f Fg(2)f(Q)j Fj(suc)m(h)h(that)e Fh(a)47 b(=)-61 b Fg(2)35 b Fh(Q)p Fj(.)i(Then)g Fh(Q)g Fj(con)m(tains)g Fh(d)d Fg(2)h Fh(E)166 913 y Fj(and)j(\(sa)m(y\))h Fh(b)f Fg(2)g Fh(T)14 b Fj(,)38 b(but)g(in)g(this)f(case)j(there)e(is)g (a)g(triangle)e Fg(f)p Fh(a;)17 b(b;)g(d)p Fg(g)38 b Fj(whic)m(h)h(con)m(tains)166 1034 y Fh(E)6 b Fj(,)31 b(con)m(tradicting)f(that)h Fh(E)37 b Fj(is)30 b(a)h(clique.)g(Hence)h (all)d(the)j(elemen)m(ts)f(of)g Fg(Q)g Fj(are)g(triangles.)166 1154 y(Since)i Fg(Q)g Fj(is)f(not)g(a)g(star,)h(there)g(m)m(ust)g(b)s (e)g(more)f(that)g(one)h(triangle)e(in)h Fg(Q)p Fj(.)166 1383 y(F)-8 b(urthermore,)38 b(w)m(e)h(claim)c(that)j(there)h(m)m(ust)f (exist)g(t)m(w)m(o)h(triangles)e(in)g Fg(Q)h 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Fj(,)f(but)h(then)g Fh(T)2914 2000 y Ft(2)2981 1985 y Fj(w)m(ould)f(ha)m(v)m(e)166 2105 y(to)33 b(meet)h Fh(T)47 b Fj(and)33 b Fh(T)877 2120 y Ft(1)950 2105 y Fj(in)g(the)h(edges)h Fg(f)p Fh(a;)17 b(c)p Fg(g)33 b Fj(and)g Fg(f)p Fh(a;)17 b(c)2143 2069 y Fe(0)2166 2105 y Fg(g)33 b Fj(and)h(therefore)g Fh(T)2907 2120 y Ft(2)2976 2105 y Fj(=)29 b Fg(f)p Fh(a;)17 b(c;)g(c)3354 2069 y Fe(0)3377 2105 y Fg(g)p Fj(.)166 2225 y(This)33 b(is)f(a)g(con)m(tradiction)g(b)s(ecause)i Fh(G)e Fj(can)h(not)f(con)m (tain)g(a)h(tetrahedron.)166 2454 y(W)-8 b(e)31 b(can)g(then)h(tak)m(e) g Fh(T)41 b Fj(=)28 b Fg(f)p Fh(a;)17 b(b;)g(c)p Fg(g)30 b Fj(so)h(that)g(there)g(is)g(a)f(triangle)f Fh(T)2658 2469 y Ft(1)2726 2454 y Fg(2)f(Q)j Fj(whic)m(h)g(meets)166 2574 y Fh(T)49 b Fj(in)35 b(just)i(one)f(v)m(ertex,)h(sa)m(y)g Fh(T)1322 2589 y Ft(1)1395 2574 y Fj(=)c Fg(f)p Fh(a;)17 b(b)1690 2538 y Fe(0)1713 2574 y Fh(;)g(c)1799 2538 y Fe(0)1822 2574 y Fg(g)p Fj(.)36 b(Since)f Fg(Q)i Fj(is)e(a)g(nec)m (ktie,)i(there)g(m)m(ust)e(b)s(e)166 2695 y(a)f(triangle)f Fh(T)666 2710 y Ft(2)736 2695 y Fg(2)e(Q)k Fj(whic)m(h)f(do)s(es)h(not) f(con)m(tain)g Fh(a)p Fj(.)h(The)g(triangle)e Fh(T)2704 2710 y Ft(2)2778 2695 y Fj(m)m(ust)h(con)m(tain)g(at)166 2815 y(least)d(one)h(v)m(ertex)h(from)d(eac)m(h)j(of)e Fh(T)45 b Fj(and)32 b Fh(T)1771 2830 y Ft(1)1810 2815 y Fj(,)g(but)g(it)e(can)i(not)f(con)m(tain)g(an)h(edge)g(of)f(an)m(y) 166 2935 y(of)40 b(these)i(triangles:)d Fg(f)p Fh(b;)17 b(c)p Fg(g)40 b(\032)i Fh(T)1417 2950 y Ft(2)1497 2935 y Fj(or)e Fg(f)p Fh(b)1715 2899 y Fe(0)1738 2935 y Fh(;)17 b(c)1824 2899 y Fe(0)1847 2935 y Fg(g)41 b(\032)g Fh(T)2113 2950 y Ft(2)2193 2935 y Fj(w)m(ould)g(imply)d(that)i Fh(G)h Fj(con)m(tains)166 3056 y(a)33 b(tetrahedron.)i(Without)e(loss)g (of)g(generalit)m(y)g(w)m(e)i(can)f(assume)g(that)g Fh(T)2884 3071 y Ft(2)2953 3056 y Fj(=)29 b Fg(f)p Fh(a)3159 3020 y Fe(0)3183 3056 y Fh(;)17 b(b;)g(c)3354 3020 y Fe(0)3377 3056 y Fg(g)p Fj(,)166 3176 y(where)24 b(the)e(v)m(ertex)i Fh(a)931 3140 y Fe(0)977 3176 y Fj(is)e(not)g(in)g Fh(T)15 b Fg([)q Fh(T)1528 3191 y Ft(1)1568 3176 y Fj(.)22 b(No)m(w)h(the)g (triangle)d Fh(T)2391 3191 y Ft(3)2458 3176 y Fj(=)28 b Fg(f)p Fh(a;)17 b(b;)g(c)2834 3140 y Fe(0)2857 3176 y Fg(g)22 b Fj(is)g(an)g(in)m(terior)166 3296 y(triangle)28 b(of)h Fh(G)p Fj(.)h(Supp)s(osing)f(that)g Fh(T)1494 3311 y Ft(3)1573 3296 y Fh(=)-60 b Fg(2)28 b(Q)p Fj(,)i(w)m(e)g(will)e (\014nd)i(a)f(con)m(tradiction.)g(Under)h(this)166 3417 y(supp)s(osition,)37 b(there)i(m)m(ust)f(exist)g(a)g(triangle)e Fh(T)1962 3432 y Ft(4)2039 3417 y Fg(2)h(Q)i Fj(suc)m(h)g(that)f Fh(T)2761 3432 y Ft(4)2826 3417 y Fg(\\)26 b Fh(T)2975 3432 y Ft(3)3052 3417 y Fj(=)37 b Fd(?)p Fj(,)h(but)166 3537 y(then)k(w)m(e)h(w)m(ould)e(ha)m(v)m(e)i Fh(T)1134 3552 y Ft(4)1201 3537 y Fg(\\)29 b Fh(T)56 b Fj(=)43 b Fg(f)p Fh(c)p Fg(g)p Fj(,)e Fh(T)1795 3552 y Ft(4)1863 3537 y Fg(\\)28 b Fh(T)2014 3552 y Ft(1)2097 3537 y Fj(=)42 b Fg(f)p Fh(b)2306 3501 y Fe(0)2330 3537 y Fg(g)p Fj(,)f(and)h Fh(T)2704 3552 y Ft(4)2771 3537 y Fg(\\)29 b Fh(T)2923 3552 y Ft(2)3005 3537 y Fj(=)43 b Fg(f)p Fh(a)3225 3501 y Fe(0)3248 3537 y Fg(g)p Fj(,)e(so)166 3658 y Fh(T)223 3673 y Ft(4)306 3658 y Fj(=)i Fg(f)p Fh(a)526 3621 y Fe(0)549 3658 y Fh(;)17 b(b)634 3621 y Fe(0)658 3658 y Fh(;)g(c)p Fg(g)41 b Fj(and)h Fh(G)f Fj(w)m(ould)h(con)m(tain)f(an)h (o)s(ctahedron,)g(whose)h(lo)s(cal)c(girth)i(is)g(4.)166 3778 y(Therefore,)36 b Fh(T)694 3793 y Ft(3)765 3778 y Fg(2)d(Q)p Fj(,)i(and)g(it)f(is)g(no)m(w)i(easy)g(to)e(see)j(that)d Fg(Q)e Fj(=)g Fg(Q)2619 3793 y Fn(T)2660 3802 y Fc(3)2699 3778 y Fj(:)j(no)g(triangle)e(of)h Fg(Q)166 3898 y Fj(can)d(meet)g Fh(T)637 3913 y Ft(3)708 3898 y Fj(in)f(just)h(one)g(v)m(ertex)i(b)s (ecause)f Fh(G)f Fj(do)s(es)h(not)e(con)m(tain)h(tetrahedra,)g(and)g Fg(Q)166 4019 y Fj(m)m(ust)i(therefore)g(con)m(tain)f(all)f(the)i (triangles)e(of)h Fh(G)g Fj(that)h(meet)f Fh(T)2588 4034 y Ft(3)2660 4019 y Fj(in)g(some)g(edge.)99 b Ff(2)166 4429 y Fi(3.2)g(R)-5 b(e)g(duction)35 b(and)f(Pr)-5 b(o)g(of)35 b(of)f(The)-5 b(or)g(em)34 b(8)166 4778 y Fj(Let)43 b(us)g(assume)g (that)g(the)g(graph)f Fh(G)h Fj(satis\014es)g(lg\()p Fh(G)p Fj(\))i Fg(\025)g Fj(7.)e(Let)g Fg(F)52 b Fj(b)s(e)43 b(a)f(family)e(of)166 4898 y(cliques)28 b(of)g Fh(k)s(G)h Fj(suc)m(h)h(that)e(an)m(y)h(t)m(w)m(o)g(mem)m(b)s(ers)f(of)g Fg(F)38 b Fj(ha)m(v)m(e)30 b(a)e(non-empt)m(y)h(in)m(tersection.)166 5019 y(In)43 b(order)f(to)h(pro)m(v)m(e)g(Theorem)g(8,)f(w)m(e)i(need)f (to)f(pro)m(v)m(e)i(that)e(the)h(in)m(tersection)g(of)f(all)166 5139 y(the)32 b(mem)m(b)s(ers)f(of)g Fg(F)40 b Fj(is)31 b(non-empt)m(y)-8 b(.)31 b(F)-8 b(or)31 b(instance,)h(if)e(ev)m(ery)j (mem)m(b)s(er)d(of)h Fg(F)41 b Fj(is)31 b(a)f(star,)166 5259 y(w)m(e)36 b(kno)m(w)h(b)m(y)f(Lemma)e(9)g(that)h(this)g(is)g (true.)h(Th)m(us)g(w)m(e)h(ma)m(y)e(as)g(w)m(ell)g(assume)g(that)g Fg(F)166 5380 y Fj(con)m(tains)40 b(some)h(nec)m(kties.)g(By)g(Prop)s 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b(ha)m(v)m(e)h(remark) m(ed)f(already)e(that)i(t)m(w)m(o)g(stars)g Fh(u)2072 869 y Fe(\003)2151 906 y Fj(and)f Fh(v)2399 869 y Fe(\003)2479 906 y Fj(in)m(tersect)h(if)e(and)i(only)f(if)166 1026 y(their)32 b(cen)m(ters)i Fh(u)f Fj(and)f Fh(v)k Fj(are)d(neigh)m(b)s (ours)g(in)f Fh(G)p Fj(.)166 1247 y(A)d(star)g Fh(u)516 1211 y Fe(\003)584 1247 y Fj(and)g(a)g(nec)m(ktie)h Fg(Q)1256 1262 y Fn(T)1341 1247 y Fj(in)e Fh(k)s(G)h Fj(ha)m(v)m(e)h(a)f (non-empt)m(y)g(in)m(tersection)g(if)f(and)h(only)g(if)166 1367 y(they)37 b(share)f(some)g(triangle)e(of)h Fh(G)p Fj(.)h(There)h(are)f(t)m(w)m(o)h(p)s(ossibilities:)c(either)i(this)h (shared)166 1487 y(triangle)i(is)i(the)g(cen)m(ter)i Fh(T)54 b Fj(of)39 b Fg(Q)1420 1502 y Fn(T)1516 1487 y Fj(\(if)f(and)j(only)e(if)g Fh(u)h Fj(is)f(a)h(v)m(ertex)i(of)e Fh(T)14 b Fj(\),)39 b(or)h Fh(u)52 b(=)-61 b Fg(2)41 b Fh(T)166 1608 y Fj(and)35 b(then)h(the)f(shared)h(triangle)d(is)h(an) h(ear)g(of)f Fg(Q)2022 1623 y Fn(T)2112 1608 y Fj(\(if)g(and)h(only)g (if)e Fh(u)43 b(=)-61 b Fg(2)32 b Fh(T)49 b Fj(but)35 b Fh(u)g Fj(is)f(a)166 1728 y(neigh)m(b)s(our)e(of)g(t)m(w)m(o)i(v)m (ertices)f(of)g Fh(T)46 b Fj(\).)166 1949 y(Tw)m(o)28 b(di\013eren)m(t)e(nec)m(kties)j Fg(Q)1203 1964 y Fn(T)1285 1949 y Fj(and)d Fg(Q)1549 1964 y Fn(T)1600 1945 y Fb(0)1654 1949 y Fj(in)g Fh(k)s(G)g Fj(ha)m(v)m(e)i(a)f(non-empt)m(y)f(in)m (tersection)h(if)f(and)166 2069 y(only)k(if)g(they)h(share)h(some)e (triangle)f(of)h Fh(G)p Fj(.)h(There)h(are)e(again)g(t)m(w)m(o)h(p)s (ossibilities:)d(either)166 2190 y(this)f(shared)i(triangle)c(is)i(the) i(cen)m(ter)f(of)g(one)f(of)g(the)h(nec)m(kties)i(and)d(an)h(ear)f(of)g (the)h(other,)166 2310 y(or)39 b(the)g(shared)h(triangle)c(is)j(an)g (ear)f(of)h(eac)m(h)h(of)e(the)h(t)m(w)m(o)h(nec)m(kties.)g(Notice)f (that)f(the)166 2430 y(\014rst)31 b(case)g(tak)m(es)h(place)e (precisely)g(when)i(the)e(cen)m(ter)i(of)e(eac)m(h)h(nec)m(ktie)g(is)f (an)g(ear)g(of)g(the)166 2551 y(other,)k(and)h(that)e(this)h(happ)s (ens)h(if)e(and)h(only)g(if)f(the)h(t)m(w)m(o)h(cen)m(ters)h Fh(T)48 b Fj(and)34 b Fh(T)3070 2515 y Fe(0)3127 2551 y Fj(meet)g(at)166 2671 y(an)c(edge.)g(The)h(second)g(case)g(tak)m(es)g (place)e(precisely)h(when)h(the)f(t)m(w)m(o)h(cen)m(ters)g Fh(T)44 b Fj(and)30 b Fh(T)3431 2635 y Fe(0)166 2792 y Fj(meet)h(at)g(a)f(v)m(ertex)j(\(sa)m(y)f Fh(T)41 b Fj(=)28 b Fg(f)p Fh(a;)17 b(b;)g(v)t Fg(g)30 b Fj(and)h Fh(T)1921 2755 y Fe(0)1972 2792 y Fj(=)d Fg(f)p Fh(a)2177 2755 y Fe(0)2200 2792 y Fh(;)17 b(b)2285 2755 y Fe(0)2308 2792 y Fh(;)g(v)t Fg(g)p Fj(\))31 b(and)g(there)g(is)g(an)g(edge)h Fh(e)166 2912 y Fj(in)27 b Fh(G)h Fj(joining)e(some)h(v)m(ertex)j(in)d (the)h(edge)h Fh(T)d Fg(\000)12 b Fh(v)32 b Fj(with)c(some)f(v)m(ertex) j(in)d(the)h(edge)h Fh(T)3278 2876 y Fe(0)3313 2912 y Fg(\000)12 b Fh(v)166 3032 y Fj(\(sa)m(y)39 b Fh(e)e Fj(=)g Fg(f)p Fh(a;)17 b(a)769 2996 y Fe(0)792 3032 y Fg(g)p Fj(\).)38 b(W)-8 b(e)38 b(sa)m(y)h(that)f Fh(e)g Fj(is)g(a)g Fi(cr)-5 b(ossb)g(ar)47 b Fj(joining)36 b Fh(T)52 b Fj(and)38 b Fh(T)2875 2996 y Fe(0)2898 3032 y Fj(.)g(Notice)g(that)166 3153 y(crossbars,)30 b(when)f(they)g(exist,) f(are)h(unique:)f(indeed,)h Fh(G)f Fj(con)m(tains)g(no)g(tetrahedron)h (and)166 3273 y(no)k(v)m(ertex)h(of)e Fh(G)g Fj(can)h(ha)m(v)m(e)h(a)e (cycle)i(of)e(length)g(4)g(in)g(its)g(neigh)m(b)s(ourho)s(o)s(d.)166 3494 y(In)27 b(view)g(of)g(these)h(remarks,)f(w)m(e)h(can)f(represen)m (t)h(the)g(stars)f(and)g(nec)m(kties)h(in)e Fg(F)36 b Fj(b)m(y)28 b(their)166 3614 y(cen)m(ters)j(and)e(th)m(us)h(Theorem)g (8)f(will)e(b)s(e)i(a)g(consequence)j(of)d(the)g(follo)m(wing)e (result.)i(The)166 3735 y(h)m(yp)s(othesis)36 b(are)g(the)f(conditions) g(that)g(the)g(cen)m(ters)i(of)e(the)h(stars)g(and)f(nec)m(kties)i(in)d Fg(F)166 3855 y Fj(m)m(ust)28 b(satisfy)g(if)e(an)m(y)j(t)m(w)m(o)f(of) f(these)i(cliques)f(in)m(tersect,)h(and)e(the)i(conclusion)e (translates)166 3975 y(the)g(fact)e(that)h(the)h(triangle)d Fh(X)34 b Fj(is)25 b(in)h(an)m(y)g(of)g(these)h(cliques.)f(W)-8 b(e)27 b(ha)m(v)m(e)g(remark)m(ed)g(ab)s(o)m(v)m(e)166 4096 y(that)32 b(w)m(e)i(can)f(assume)g(that)f Fg(F)42 b Fj(con)m(tains)33 b(some)f(nec)m(ktie.)166 4316 y Fk(Theorem)37 b(11)49 b Fi(Assume)40 b(that)h Fj(lg\()p Fh(G)p Fj(\))c Fg(\025)h Fj(7)i Fi(and)f(let)i Fg(F)49 b Fi(b)-5 b(e)40 b(a)g(family)g(of)g(vertic)-5 b(es)39 b(and)166 4437 y(triangles)e(of)h Fh(G)g Fi(which)f(includes)g(at)i(le)-5 b(ast)37 b(one)h(triangle)f(and)h(satis\014es)f(the)h(fol)5 b(lowing)166 4557 y(thr)-5 b(e)g(e)35 b(c)-5 b(onditions:)213 4778 y(\(i\))49 b(A)n(ny)35 b(two)g(vertic)-5 b(es)34 b(in)h Fg(F)44 b Fi(ar)-5 b(e)35 b(neighb)-5 b(ours.)183 4898 y(\(ii\))49 b(F)-7 b(or)49 b(e)-5 b(ach)49 b(triangle)g(and)g(e)-5 b(ach)49 b(vertex)h(in)f Fg(F)10 b Fi(,)49 b(either)h(the)g(vertex)f (lies)g(on)h(the)372 5019 y(triangle)34 b(or)h(it)g(is)g(a)g(neighb)-5 b(our)34 b(of)g(two)h(vertic)-5 b(es)35 b(of)f(the)h(triangle.)153 5139 y(\(iii\))49 b(Given)40 b(two)h(di\013er)-5 b(ent)40 b(triangles)f(in)i Fg(F)10 b Fi(,)39 b(either)i(they)g(me)-5 b(et)40 b(in)g(an)g(e)-5 b(dge)40 b(or)g(they)372 5259 y(me)-5 b(et)35 b(in)f(just)h(one)g(vertex)f(and)h(they)g(ar)-5 b(e)35 b(joine)-5 b(d)34 b(by)h(a)f(cr)-5 b(ossb)g(ar.)472 5380 y(Then)34 b(ther)-5 b(e)35 b(exists)f(a)h(triangle)f Fh(X)43 b Fi(in)34 b Fh(G)h Fi(such)g(that:)1769 5712 y Fj(8)p eop %%Page: 9 9 9 8 bop 170 83 a Fi(\(A\))50 b(Each)34 b(triangle)h(of)f Fg(F)45 b Fi(me)-5 b(ets)34 b Fh(X)43 b Fi(in)34 b(an)h(e)-5 b(dge)34 b(or)h(is)g(e)-5 b(qual)34 b(to)h Fh(X)8 b Fi(,)35 b(and)174 203 y(\(B\))49 b(e)-5 b(ach)34 b(vertex)h(of)f Fg(F)45 b Fi(is)34 b(in)h Fh(X)8 b Fi(.)166 693 y Fk(PR)m(OOF.)49 b Fj(Let)38 b(us)h(study)h(\014rst)f(the)g(case)g(in)f(whic)m(h)h(an)m (y)g(t)m(w)m(o)g(triangles)e(in)h Fg(F)48 b Fj(meet)166 813 y(in)35 b(an)g(edge.)h(Let)f Fh(T)46 b Fj(=)32 b Fg(f)p Fh(a;)17 b(b;)g(c)p Fg(g)35 b Fj(b)s(e)h(an)m(y)f(triangle)f(in) g Fg(F)10 b Fj(.)35 b(If)g(all)f(the)h(v)m(ertices)i(of)d Fg(F)45 b Fj(lie)166 934 y(in)33 b Fh(T)14 b Fj(,)34 b(then)g Fh(X)k Fj(=)29 b Fh(T)48 b Fj(will)31 b(clearly)i(do.)h(Then)h (w)m(e)g(can)f(assume)g(that)g(there)g(is)f(a)h(v)m(ertex)166 1054 y Fh(u)40 b Fg(2)i(F)50 b Fj(suc)m(h)41 b(that)f Fh(u)52 b(=)-60 b Fg(2)41 b Fh(T)14 b Fj(.)40 b(By)h(\(ii\))d(w)m(e)k (can)e(assume)h(that)f Fg(f)p Fh(u;)17 b(a)p Fg(g)p Fh(;)g Fg(f)p Fh(u;)g(b)p Fg(g)39 b(2)j Fh(E)6 b Fj(\()p Fh(G)p Fj(\).)166 1174 y(Notice)38 b(that)h Fh(u)f Fj(is)h(the)g(only)f(v)m (ertex)j(of)d Fg(F)49 b Fj(whic)m(h)39 b(is)g(not)f(in)g Fh(T)14 b Fj(:)39 b(Indeed)h(if)e Fh(v)k Fg(2)d(F)48 b Fj(is)166 1295 y(not)38 b(in)g Fh(T)52 b Fj(and)38 b Fh(u)f Fg(6)p Fj(=)h Fh(v)t Fj(,)g(w)m(e)h(ha)m(v)m(e)h Fg(f)p Fh(u;)17 b(v)t Fg(g)36 b(2)i Fh(E)6 b Fj(\()p Fh(G)p Fj(\))38 b(b)m(y)h(\(i\),)f(and)g(b)m(y)h(\(ii\))e(there)i(w)m (ould)166 1415 y(b)s(e)i(t)m(w)m(o)g(p)s(ossibilities:)d(either)i Fg(f)p Fh(v)t(;)17 b(a)p Fg(g)p Fh(;)g Fg(f)p Fh(v)t(;)g(b)p Fg(g)40 b(2)h Fh(E)6 b Fj(\()p Fh(G)p Fj(\),)41 b(but)f(then)i Fg(f)p Fh(a;)17 b(b;)g(u;)g(v)t Fg(g)39 b Fj(w)m(ould)166 1536 y(b)s(e)k(a)f(tetrahedron,)h(or)f(\(sa)m(y\))h Fg(f)p Fh(v)t(;)17 b(b)p Fg(g)p Fh(;)g Fg(f)p Fh(v)t(;)g(c)p Fg(g)43 b(2)i Fh(E)6 b Fj(\()p Fh(G)p Fj(\),)42 b(but)h(then)g Fh(c;)17 b(a;)g(u;)g(v)t(;)g(c)41 b Fj(w)m(ould)166 1656 y(b)s(e)d(a)g(cycle)g(of)g(length)f(four)h(in)f Fh(N)10 b Fj(\()p Fh(b)p Fj(\).)39 b(Notice)e(also)g(that)h Fh(c)48 b(=)-61 b Fg(2)38 b(F)47 b Fj(b)s(ecause)39 b(otherwise)166 1776 y Fg(f)p Fh(a;)17 b(b;)g(c;)g(u)p Fg(g)23 b Fj(w)m(ould)i(b)s(e)f (a)g(tetrahedron)h(b)m(y)h(\(i\).)d(Therefore,)j Fh(X)35 b Fj(=)28 b Fg(f)p Fh(u;)17 b(a;)g(b)p Fg(g)23 b Fj(satis\014es)j (\(B\).)166 1897 y(W)-8 b(e)25 b(pro)m(v)m(e)i(\(A\))e(b)m(y)g(con)m (tradiction:)f(Let)i Fh(T)1735 1861 y Fe(0)1783 1897 y Fj(b)s(e)f(a)f(triangle)f(of)i Fg(F)34 b Fj(suc)m(h)27 b(that)e Fh(T)3029 1861 y Fe(0)3079 1897 y Fg(6)p Fj(=)j Fh(X)33 b Fj(and)166 2017 y Fh(T)237 1981 y Fe(0)294 2017 y Fj(do)s(es)h(not)g(meet)g Fh(X)42 b Fj(in)33 b(an)h(edge.)g (Then)h Fh(T)1882 1981 y Fe(0)1939 2017 y Fj(m)m(ust)f(meet)g Fh(T)48 b Fj(in)33 b(an)h(edge)h(other)f(than)166 2138 y Fg(f)p Fh(a;)17 b(b)p Fg(g)p Fj(,)31 b(sa)m(y)h Fg(f)p Fh(a;)17 b(c)p Fg(g)p Fj(,)30 b(so)h Fh(T)1110 2101 y Fe(0)1161 2138 y Fj(=)c Fg(f)p Fh(a;)17 b(c;)g(b)1536 2101 y Fe(0)1559 2138 y Fg(g)p Fj(.)31 b(By)g(\(ii\))e(w)m(e)j(m)m(ust) f(ha)m(v)m(e)h(either)f(an)f(edge)i Fg(f)p Fh(u;)17 b(c)p Fg(g)166 2258 y Fj(or)31 b(an)f(edge)i Fg(f)p Fh(u;)17 b(b)829 2222 y Fe(0)852 2258 y Fg(g)p Fj(,)31 b(but)g Fg(f)p Fh(u;)17 b(c)p Fg(g)26 b(2)i Fh(E)6 b Fj(\()p Fh(G)p Fj(\))31 b(implies)e(that)h Fg(f)p Fh(a;)17 b(b;)g(c;)g(u)p Fg(g)30 b Fj(is)g(a)h(tetrahedron,)166 2378 y(and)36 b Fg(f)p Fh(u;)17 b(b)550 2342 y Fe(0)573 2378 y Fg(g)33 b(2)h Fh(E)6 b Fj(\()p Fh(G)p Fj(\))35 b(implies)f(that)h Fh(u;)17 b(b;)g(c;)g(b)1883 2342 y Fe(0)1906 2378 y Fh(;)g(u)35 b Fj(is)h(a)f(cycle)i(of)e(length)g(four)h(in)f Fh(N)10 b Fj(\()p Fh(a)p Fj(\).)166 2499 y(Therefore,)34 b Fh(X)h Fj(=)28 b Fg(f)p Fh(u;)17 b(a;)g(b)p Fg(g)32 b Fj(also)f(satis\014es)j (\(A\).)166 2737 y(No)m(w)k(w)m(e)g(can)f(assume)h(that)e(there)i(are)f (t)m(w)m(o)h(triangles)d(in)i Fg(F)46 b Fj(whic)m(h)38 b(do)f(not)g(meet)f(in)166 2857 y(an)41 b(edge:)g(b)m(y)h(\(iii\))c (they)j(meet)g(in)f(one)h(v)m(ertex)i(and)e(are)f(joined)h(b)m(y)g(a)g (crossbar.)g(Let)166 2978 y Fh(T)223 2993 y Ft(1)293 2978 y Fj(=)31 b Fg(f)p Fh(x;)17 b(b;)g(a)p Fg(g)35 b Fj(and)f Fh(T)1018 2993 y Ft(2)1088 2978 y Fj(=)d Fg(f)p Fh(x;)17 b(d;)g(c)p Fg(g)33 b Fj(b)s(e)i(these)h(t)m(w)m(o)f(triangles) e(in)g Fg(F)10 b Fj(.)34 b(Without)g(loss)g(of)166 3098 y(generalit)m(y)h(w)m(e)j(assume)e(that)g Fg(f)p Fh(b;)17 b(c)p Fg(g)36 b Fj(is)g(the)h(crossbar)g(joining)d Fh(T)2626 3113 y Ft(1)2701 3098 y Fj(and)j Fh(T)2952 3113 y Ft(2)2991 3098 y Fj(.)g(W)-8 b(e)36 b(claim)166 3218 y(that)c(the)h(triangle)e Fh(X)36 b Fj(=)27 b Fg(f)p Fh(x;)17 b(c;)g(b)p Fg(g)33 b Fj(satis\014es)g(\(A\))f(and)h(\(B\).)166 3456 y(The)e(pro)s(of)f(of) g(\(A\))g(will)e(b)s(e)j(indirect:)e(assuming)h(that)g(there)h(exists)g (a)f(triangle)f Fh(T)41 b Fg(2)28 b(F)166 3577 y Fj(that)j(do)s(es)h (not)f(meet)g Fh(X)39 b Fj(in)31 b(an)g(edge)h(and)g Fh(T)45 b Fj(is)31 b(not)g Fh(X)8 b Fj(,)31 b(w)m(e)h(will)d(get)j(a)f (con)m(tradiction.)166 3815 y(First)36 b(w)m(e)h(claim)e(that)h Fh(x)47 b(=)-61 b Fg(2)35 b Fh(T)14 b Fj(:)36 b(Assuming)h(that)f Fh(x)f Fg(2)g Fh(T)14 b Fj(,)37 b(sa)m(y)g Fh(T)48 b Fj(=)35 b Fg(f)p Fh(x;)17 b(u;)g(v)t Fg(g)p Fj(,)35 b(w)m(e)j(will)166 3935 y(get)c(a)f(con)m(tradiction.)g(T)-8 b(o)33 b(b)s(egin)g(with,)h Fh(T)47 b Fj(m)m(ust)34 b(meet)f(at)h(least)f(one)h(of)f Fh(T)2978 3950 y Ft(1)3051 3935 y Fj(and)h Fh(T)3299 3950 y Ft(2)3372 3935 y Fj(in)166 4056 y(an)f(edge:)g(Indeed,)i(if)d Fh(T)46 b Fj(meets)33 b(b)s(oth)g Fh(T)1654 4071 y Ft(1)1727 4056 y Fj(and)g Fh(T)1974 4071 y Ft(2)2046 4056 y Fj(in)f(just)h(the)h (v)m(ertex)g Fh(x)p Fj(,)g(w)m(e)g(ha)m(v)m(e)g(t)m(w)m(o)166 4176 y(crossbars)43 b(joining)c Fh(T)55 b Fj(with)41 b Fh(T)1325 4191 y Ft(1)1406 4176 y Fj(and)g Fh(T)1661 4191 y Ft(2)1701 4176 y Fj(,)g(i.e.,)g(joining)e Fg(f)p Fh(u;)17 b(v)t Fg(g)40 b Fj(with)h Fg(f)p Fh(a;)17 b(b)p Fg(g)41 b Fj(and)h(also)166 4296 y(with)29 b Fg(f)p Fh(c;)17 b(d)p Fg(g)p Fj(;)28 b(but)i(this)f(implies)e(the)j(existence)h(in)d Fh(N)10 b Fj(\()p Fh(x)p Fj(\))31 b(of)d(a)h(cycle)h(of)f(length)g(at)g (most)166 4417 y(6.)g(W)-8 b(e)30 b(can)f(therefore)h(assume)g(that)f Fh(T)43 b Fj(meets)29 b Fh(T)1989 4432 y Ft(1)2058 4417 y Fj(in)f(some)h(edge)h(and,)g(since)f(this)g(edge)166 4537 y(cannot)39 b(b)s(e)f Fg(f)p Fh(x;)17 b(b)p Fg(g)38 b(\032)g Fh(X)8 b Fj(,)38 b(it)g(m)m(ust)g(b)s(e)h Fg(f)p Fh(x;)17 b(a)p Fg(g)38 b Fj(and)g(so)h Fh(T)51 b Fj(=)38 b Fg(f)p Fh(x;)17 b(a;)g(v)t Fg(g)p Fj(.)38 b(No)m(w)g Fh(T)53 b Fj(m)m(ust)166 4658 y(also)39 b(meet)h Fh(T)672 4673 y Ft(2)751 4658 y Fj(in)f(an)h(edge:)h(Indeed,)g(if)e Fh(T)54 b Fj(w)m(ere)41 b(to)e(meet)h Fh(T)2492 4673 y Ft(2)2572 4658 y Fj(in)f(just)h(the)g(v)m(ertex)i Fh(x)p Fj(,)166 4778 y(there)35 b(w)m(ould)f(exist)g(a)f(crossbar)i(joining)d Fg(f)p Fh(a;)17 b(v)t Fg(g)33 b Fj(with)h Fg(f)p Fh(c;)17 b(d)p Fg(g)p Fj(,)33 b(but)h(since)g Fh(v)t(;)17 b(a;)g(b;)g(c;)g(d)33 b Fj(is)166 4898 y(a)g(path)h(in)f Fh(N)10 b Fj(\()p Fh(x)p Fj(\),)34 b(it)f(w)m(ould)g(follo)m(w)f(that)h(there)h(exists)h (in)d Fh(N)10 b Fj(\()p Fh(x)p Fj(\))35 b(of)e(a)g(cycle)h(of)f(length) 166 5019 y(at)h(most)g(5.)g(Therefore,)h Fh(T)48 b Fj(meets)35 b Fh(T)1549 5034 y Ft(2)1622 5019 y Fj(in)f(an)g(edge)h(whic)m(h)g(con) m(tains)f Fh(x)p Fj(,)h(and)f(this)g(edge)166 5139 y(cannot)27 b(b)s(e)f Fg(f)p Fh(x;)17 b(c)p Fg(g)28 b(\032)g Fh(X)8 b Fj(,)26 b(so)h Fh(T)40 b Fj(meets)27 b Fh(T)1661 5154 y Ft(2)1727 5139 y Fj(in)e(the)i(edge)g Fg(f)p Fh(x;)17 b(d)p Fg(g)26 b Fj(and)h(th)m(us)g Fh(T)42 b Fj(=)27 b Fg(f)p Fh(x;)17 b(a;)g(d)p Fg(g)p Fj(.)166 5259 y(No)m(w)41 b Fh(a;)17 b(b;)g(c;)g(d;)g(a)39 b Fj(is)g(a)h(cycle)h(of)e(length)h(4) f(in)h Fh(N)10 b Fj(\()p Fh(x)p Fj(\),)40 b(and)h(this)e(con)m (tradiction)g(sho)m(ws)166 5380 y(that)32 b Fh(x)40 b(=)-60 b Fg(2)28 b Fh(T)14 b Fj(.)1769 5712 y(9)p eop %%Page: 10 10 10 9 bop 166 83 a Fj(No)m(w)47 b(that)f(w)m(e)h(kno)m(w)g(that)f Fh(x)63 b(=)-61 b Fg(2)52 b Fh(T)14 b Fj(,)46 b(w)m(e)h(get)f(that)g Fh(T)60 b Fj(m)m(ust)46 b(con)m(tain)g(at)g(least)f(one)166 203 y(v)m(ertex)c(from)c Fg(f)p Fh(a;)17 b(b)p Fg(g)39 b Fj(and)g(at)f(least)h(one)g(from)f Fg(f)p Fh(c;)17 b(d)p Fg(g)p Fj(.)37 b(But)i(then)h(these)g(t)m(w)m(o)f(v)m(ertices)166 324 y(are)25 b(neigh)m(b)s(ours)g(and)g(therefore)h(the)f(edge)h (joining)d(them)i(is)f(a)h(crossbar)h(joining)d Fh(T)3232 339 y Ft(1)3297 324 y Fj(and)166 444 y Fh(T)223 459 y Ft(2)263 444 y Fj(.)33 b(By)i(the)f(uniqueness)h(of)f(crossbars)h(w)m (e)f(obtain)f(that)g Fg(f)p Fh(b;)17 b(c)p Fg(g)34 b Fj(is)f(an)h(edge)g(of)f Fh(T)14 b Fj(,)34 b(and)166 565 y(th)m(us)28 b(either)f Fh(T)42 b Fj(=)27 b Fh(X)35 b Fj(or)27 b Fh(T)41 b Fj(meets)27 b Fh(X)35 b Fj(in)26 b(this)h(edge:)h(this)f(is)f(the)i(desired)g(con)m(tradiction,)166 685 y(and)33 b(\(A\))f(is)g(pro)m(v)m(ed.)166 909 y(No)m(w)k(w)m(e)g (pro)m(v)m(e)h(\(B\):)e(Let)h Fh(v)j Fj(b)s(e)c(a)g(v)m(ertex)i(of)e Fg(F)45 b Fj(and)35 b(supp)s(ose)i(that)e Fh(v)48 b(=)-61 b Fg(2)33 b Fh(X)8 b Fj(.)35 b(W)-8 b(e)36 b(will)166 1029 y(get)h(a)f(con)m(tradiction.)g(If)g Fh(v)j Fg(2)c Fh(T)1388 1044 y Ft(1)1453 1029 y Fg([)25 b Fh(T)1601 1044 y Ft(2)1640 1029 y Fj(,)37 b(then)h Fh(v)i Fj(m)m(ust)d(b)s(e)g (either)f Fh(a)h Fj(or)f Fh(d)p Fj(,)h(sa)m(y)g Fh(v)i Fj(=)34 b Fh(a)p Fj(.)166 1150 y(By)41 b(\(ii\),)e(w)m(e)j(m)m(ust)f (ha)m(v)m(e)h(either)e Fg(f)p Fh(a;)17 b(c)p Fg(g)41 b(2)h Fh(E)6 b Fj(\()p Fh(G)p Fj(\))41 b(or)f Fg(f)p Fh(a;)17 b(d)p Fg(g)41 b(2)g Fh(E)6 b Fj(\()p Fh(G)p Fj(\),)41 b(but)g(the)g(\014rst)166 1270 y(of)d(these)i(giv)m(es)g(a)e (tetrahedron)i Fg(f)p Fh(a;)17 b(b;)g(c;)g(x)p Fg(g)p Fj(,)38 b(and)h(the)g(second)h(a)f(cycle)g Fh(a;)17 b(d;)g(c;)g(b;)g(a) 38 b Fj(in)166 1391 y Fh(N)10 b Fj(\()p Fh(x)p Fj(\).)32 b(Assuming)e(that)h Fh(v)43 b(=)-60 b Fg(2)28 b Fh(T)1330 1406 y Ft(1)1388 1391 y Fg([)19 b Fh(T)1530 1406 y Ft(2)1570 1391 y Fj(,)31 b(w)m(e)h(will)d(also)h(deriv)m(e)h(a)g(con)m (tradiction.)f(If)h Fh(v)j Fj(is)d(a)166 1511 y(neigh)m(b)s(our)h(of)g Fh(x)p Fj(,)h(b)m(y)h(\(ii\))d(there)i(m)m(ust)g(b)s(e)f(one)h(edge)h (joining)c Fh(v)36 b Fj(with)c Fg(f)p Fh(a;)17 b(b)p Fg(g)28 b(\032)g Fh(T)3224 1526 y Ft(1)3297 1511 y Fj(and)166 1631 y(another)34 b(edge)g(joining)e Fh(v)38 b Fj(with)33 b Fg(f)p Fh(c;)17 b(d)p Fg(g)29 b(\032)h Fh(T)1810 1646 y Ft(2)1883 1631 y Fj(but,)35 b(since)f Fh(a;)17 b(b;)g(c;)g(d)33 b Fj(is)g(a)h(path)g(in)f Fh(N)10 b Fj(\()p Fh(x)p Fj(\),)166 1752 y(it)31 b(follo)m(ws)f(that)h(there)i(exists)f(in)f Fh(N)10 b Fj(\()p Fh(x)p Fj(\))32 b(a)g(cycle)g(of)f(length)g(at)h (most)f(5.)g(Therefore)i Fh(v)i Fj(is)166 1872 y(not)e(a)h(neigh)m(b)s (our)f(of)g Fh(x)p Fj(,)h(but)g(then)g(\(ii\))d(no)m(w)k(implies)c (that)i Fh(v)k Fj(is)c(a)h(neigh)m(b)s(our)f(of)g(b)s(oth)166 1992 y Fh(a)i Fj(and)g Fh(b)p Fj(,)g(and)g(also)f Fh(v)k Fj(is)d(a)f(neigh)m(b)s(our)g(of)h Fh(c)f Fj(and)h Fh(d)p Fj(.)g(Therefore)h Fh(v)t(;)17 b(a;)g(x;)g(c;)g(v)37 b Fj(is)e(a)f(cycle)166 2113 y(in)e Fh(N)10 b Fj(\()p Fh(b)p Fj(\),)33 b(and)g(this)f(con)m(tradiction)g(completes)g(the)h (pro)s(of.)97 b Ff(2)166 2541 y Fk(4)112 b(Whitney)37 b(triangulations)166 2885 y Fj(Let)27 b(us)f(consider)h(a)f (triangulation)d(\(i.e.,)j(simplicial)c(decomp)s(osition\))i Fg(T)52 b Fj(of)26 b(some)g(com-)166 3006 y(pact)c(\(connected\))i (surface.)f(If)f(w)m(e)h(call)d Fh(G)i Fj(the)h(underlying)e(graph)h (\(i.e.,)g(the)h(1-sk)m(eleton\))166 3126 y(of)k Fg(T)f Fj(,)h(then)h(ev)m(ery)h(2-simplex)d(of)h Fg(T)53 b Fj(is)26 b(a)i(triangle)d(\(three-v)m(ertex)k(complete)e(subgraph\))166 3246 y(of)i Fh(G)p Fj(.)f(A)h(triangulation)d(in)i(whic)m(h,)i(con)m(v) m(ersely)-8 b(,)31 b(ev)m(ery)g(triangle)c(of)h Fh(G)h Fj(is)f(a)h(face)g(of)g Fg(T)54 b Fj(is)166 3367 y(called)29 b(a)g Fi(Whitney)k(triangulation)p Fj(.)d(In)g(1931,)f(H.)h(Whitney)g (pro)m(v)m(ed)i(that)d(the)i(underly-)166 3487 y(ing)e(graph)h(of)g(an) m(y)h(suc)m(h)h(triangulation)27 b(of)i(the)i(sphere)h(is)d(a)h (Hamiltonian)d(graph)j([15].)166 3608 y(With)36 b(other)h(names,)f (Whitney)h(triangulations)d(ha)m(v)m(e)k(b)s(een)f(considered)g(also)f (b)m(y)h(W.)166 3728 y(T.)30 b(T)-8 b(utte)29 b(in)g([14])f(\(Simple)g (triangulations\))e(and)j(b)m(y)h(N.)f(Harts\014eld)g(and)g(G.)g (Ringel)e(in)166 3848 y([4])33 b(\(Clean)f(triangulations\).)166 4072 y(Whitney)45 b(triangulations)d(are)i(quite)h(amenable)e(for)h (graph-theoretical)f(considera-)166 4193 y(tions)21 b(b)s(ecause)i (they)g(are)f(determined)f(b)m(y)i(their)e(underlying)g(graph:)h(the)g (t)m(w)m(o-dimensional)166 4313 y(faces)h(are)g(just)g(the)g(triangles) 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Fj(of)e(the)i(closed)g(surface)g(triangulated)e(b)m (y)i(a)f(lo)s(cally)e(cyclic)i(graph)g Fh(G)h Fj(is)f(v)m(ery)i(easy)f (to)1745 5712 y(10)p eop %%Page: 11 11 11 10 bop 166 86 a Fj(calculate:)31 b(if)f Fh(G)i Fj(has)g Fh(n)g Fj(v)m(ertices)h(and)f(its)f(a)m(v)m(erage)i(degree)g(is)p 2484 7 51 4 v 31 w Fh(d)27 b Fg(2)h Fd(Q)12 b Fj(,)38 b(then)32 b Fh(\037)c Fj(=)3216 47 y Fn(n)p 3216 63 43 4 v 3220 120 a Ft(6)3269 86 y Fj(\(6)20 b Fg(\000)p 166 127 51 4 v 166 206 a Fh(d)p Fj(\))31 b Fg(2)h Fd(Z)p Fj(.)h(Of)h(course,)i(it)e(is)h(not)g(true)g(that)g(ev)m(ery)h(surface) g(triangulation)c(is)i(Whitney:)166 327 y(for)g(instance,)h(the)h(Hea)m (w)m(o)s(o)s(d)f(map)f(in)g(the)h(torus)g(has)g Fh(K)2335 342 y Ft(7)2409 327 y Fj(as)g(underlying)f(graph)h(and)166 447 y(this)42 b(is)h(not)f(lo)s(cally)e(cyclic.)j(Ho)m(w)m(ev)m(er,)i (the)e(class)g(of)g(Whitney)g(triangulations)d(is)i(a)166 567 y(wide)36 b(one:)g(the)g(\014rst)g(barycen)m(tric)h(sub)s(division) d(of)i(an)m(y)g(triangulation)c(of)k(a)f(compact)166 688 y(surface)e(is)g(alw)m(a)m(ys)g(Whitney)-8 b(.)166 908 y(Pro)m(ving)32 b(the)h(follo)m(wing)d(result)j(is)f(the)h(main)e (purp)s(ose)i(of)f(this)h(section:)166 1128 y Fk(Theorem)k(12)49 b Fi(If)39 b Fh(G)g Fi(is)g(a)g(lo)-5 b(c)g(al)5 b(ly)39 b(cyclic)g(gr)-5 b(aph)39 b(with)g(minimum)f(de)-5 b(gr)g(e)g(e)39 b Fh(\016)t Fj(\()p Fh(G)p Fj(\))d Fg(\025)g Fj(7)p Fi(,)166 1248 y(then)f Fh(k)437 1212 y Ft(3)476 1248 y Fh(G)581 1220 y Fg(\030)582 1252 y Fj(=)686 1248 y Fh(k)s(G)p Fi(.)166 1468 y Fj(This)29 b(will)e(b)s(e)j(pro)m(v)m(ed)g(in)f Fg(x)p Fj(4.3.)g(Notice)g(that)g(for)f(a)h(lo)s(cally)e(cyclic)i(graph) g Fh(G)g Fj(the)h(condi-)166 1588 y(tions)h(\\)p Fh(G)h Fj(is)f(regular)g(of)h(degree)h Fh(d)p Fj(")e(and)h(\\)p Fh(G)g Fj(is)f(a)h(lo)s(cally)d Fh(C)2437 1603 y Fn(d)2509 1588 y Fj(graph")j(are)g(equiv)-5 b(alen)m(t.)166 1709 y(Therefore,)25 b(the)f(ab)s(o)m(v)m(e)h(theorem)f(con)m(tains)f(the)i (follo)m(wing)c(result,)j(whic)m(h,)g(as)g(remark)m(ed)166 1829 y(in)29 b(the)h(In)m(tro)s(duction,)f(completes)g(the)h (determination)d(of)i(the)h Fh(k)s Fj(-b)s(eha)m(viour)f(of)g(all)e (the)166 1950 y(lo)s(cally)j Fh(C)545 1965 y Fn(t)607 1950 y Fj(graphs)j(for)f Fh(t)c Fg(\025)g Fj(3:)166 2170 y Fk(Theorem)37 b(13)49 b Fi(If)34 b Fh(G)h Fi(is)g(a)f(lo)-5 b(c)g(al)5 b(ly)35 b Fh(C)1572 2185 y Fn(d)1647 2170 y Fi(gr)-5 b(aph)34 b(and)g Fh(d)28 b Fg(\025)g Fj(7)p Fi(,)34 b(then)h Fh(k)2669 2133 y Ft(3)2709 2170 y Fh(G)2813 2142 y Fg(\030)2814 2174 y Fj(=)2918 2170 y Fh(k)s(G)p Fi(.)100 b Ff(2)166 2501 y Fi(4.1)f(Char)-5 b(acterization)34 b(of)g(Whitney)i(T)-7 b(riangulations)166 2841 y Fj(W)f(e)33 b(refer)g(to)f([7])h(for)f(the)h(elemen)m(ts)g(of)f(surface)h (triangulations.)166 3061 y Fk(Prop)s(osition)j(14)48 b Fi(The)32 b(gr)-5 b(aph)31 b Fh(G)h Fi(is)g(the)g Fj(1)p Fi(-skeleton)f(of)h(a)g(Whitney)g(triangulation)g(of)166 3182 y(some)i(close)-5 b(d)34 b(surfac)-5 b(e)35 b(if)f(and)g(only)h (if)g Fh(G)g Fi(is)f(a)h(lo)-5 b(c)g(al)5 b(ly)34 b(cyclic)h(gr)-5 b(aph.)166 3513 y Fk(PR)m(OOF.)49 b Fj(Let)25 b Fh(G)h Fj(b)s(e)g(the)f(1-sk)m(eleton)h(of)f(a)g(triangulation)e Fg(T)51 b Fj(of)25 b(some)g(closed)h(surface.)166 3634 y(If)38 b Fh(v)i Fg(2)d Fh(V)21 b Fj(\()p Fh(G)p Fj(\),)38 b(the)g(faces)g(around)g Fh(v)j Fj(yield)c(a)g(cycle)i Fh(C)44 b Fj(with)38 b Fh(V)21 b Fj(\()p Fh(C)7 b Fj(\))36 b(=)g Fh(N)10 b Fj(\()p Fh(v)t Fj(\).)38 b(If)g Fg(T)63 b Fj(is)166 3754 y(Whitney)26 b(this)f(cycle)h(m)m(ust)g(b)s(e)f (induced)h(b)s(ecause)h(if)d Fh(C)33 b Fj(had)25 b(a)g(c)m(hord)h(some) g(edge)g(w)m(ould)166 3874 y(b)s(elong)32 b(to)g(more)g(than)g(t)m(w)m (o)i(faces.)166 4094 y(Con)m(v)m(ersely)-8 b(,)37 b(assume)e(that)f Fh(G)h Fj(is)f(lo)s(cally)e(cyclic)j(and)f(put)h(a)g(2-cell)e(on)h(the) h(b)s(oundary)166 4215 y(of)e(eac)m(h)i(triangle.)d(Since)i(eac)m(h)h (edge)f(is)f(on)h(exactly)g(2)g(triangles,)e(eac)m(h)j(in)m(terior)e(p) s(oin)m(t)166 4335 y(in)e(an)h(edge)g(has)h(a)e(neigh)m(b)s(ourho)s(o)s (d)g(homeomorphic)f(to)i Fd(R)2395 4299 y Ft(2)2440 4335 y Fj(.)g(The)h(lo)s(cally)c(cyclic)j(con-)166 4456 y(dition)37 b(also)g(guaran)m(tees)j(that)e(eac)m(h)i(v)m(ertex)g(has)f(a)f(neigh)m (b)s(ourho)s(o)s(d)f(homeomorphic)166 4576 y(to)32 b Fd(R)351 4540 y Ft(2)397 4576 y Fj(.)97 b Ff(2)166 4808 y Fk(Prop)s(osition)36 b(15)48 b Fi(The)32 b(gr)-5 b(aph)31 b Fh(G)h Fi(is)g(the)g Fj(1)p Fi(-skeleton)f(of)h(a)g(Whitney)g (triangulation)g(of)166 4928 y(some)38 b(c)-5 b(omp)g(act)38 b(surfac)-5 b(e)38 b(if)g(and)g(only)h(if)f(the)h(op)-5 b(en)38 b(neighb)-5 b(ourho)g(o)g(d)37 b(of)h(any)h(vertex)f(of)166 5048 y Fh(G)d Fi(is)f(either)h(a)g(cycle)f(or)h(a)g(p)-5 b(ath.)166 5380 y Fk(PR)m(OOF.)49 b Fj(It)32 b(is)h(analogous)e(to)h (the)h(previous)g(pro)s(of.)97 b Ff(2)1745 5712 y Fj(11)p eop %%Page: 12 12 12 11 bop 166 83 a Fj(If)26 b Fh(G)g Fj(is)g(the)g(1-sk)m(eleton)h(of)e (some)h(Whitney)h(triangulation)c(of)i(a)h(compact)g(surface)h(with)166 203 y(b)s(order,)32 b(there)h(are)g(t)m(w)m(o)g(t)m(yp)s(es)g(of)f(v)m (ertices)h(of)f Fh(G)p Fj(:)g(The)i(v)m(ertices)f(lying)e(in)g(the)i (in)m(terior)166 324 y(of)d(the)h(surface)h(\(i.e.)e(the)h Fi(interior)42 b Fj(v)m(ertices\))31 b(ha)m(v)m(e)h(cyclic)f(op)s(en)g (neigh)m(b)s(ourho)s(o)s(d,)f(and)166 444 y(so)41 b(the)f(lo)s(cal)e (girth)h(at)h(these)i(v)m(ertices)f(is)f(their)g(degree.)h(F)-8 b(or)40 b(the)g(v)m(ertices)i(lying)d(on)166 565 y(the)c(b)s(order)f (of)f(the)i(surface)g(the)f(op)s(en)g(neigh)m(b)s(ourho)s(o)s(d)f(is)h (a)g(path,)g(and)g(so)g(the)h(lo)s(cal)166 685 y(girth)28 b(at)i(the)f Fi(exterior)40 b Fj(v)m(ertices)31 b(is)e(in\014nite.)g (Th)m(us)i(in)d(this)h(case)i(the)f(lo)s(cal)d(girth)h(is)h(the)166 805 y(minim)m(um)21 b(of)i(the)i(degrees)g(of)e(the)i(in)m(terior)d(v)m (ertices)k(\(if)c(there)j(are)f(no)g(in)m(terior)e(v)m(ertices,)166 926 y(the)35 b(lo)s(cal)e(girth)h(is)g(in\014nite\).)g(Therefore,)j (the)e(follo)m(wing)d(result)j(is)f(a)h(consequence)j(of)166 1046 y(Theorem)33 b(8:)166 1290 y Fk(Theorem)k(16)49 b Fi(If)32 b Fh(G)h Fi(is)f(the)h Fj(1)p Fi(-skeleton)f(of)g(some)g (Whitney)h(triangulation)g(of)f(a)g(c)-5 b(om-)166 1410 y(p)g(act)35 b(surfac)-5 b(e)35 b(\(with)g(or)h(without)f(b)-5 b(or)g(der\))35 b(and)g(the)g(minimum)g(de)-5 b(gr)g(e)g(e)34 b(of)i(the)f(interior)166 1531 y(vertic)-5 b(es)28 b(of)g Fh(G)g Fi(is)g(at)g(le)-5 b(ast)28 b Fj(7)p Fi(,)g(then)g Fh(k)s(G)g Fi(is)g(clique-Hel)5 b(ly)28 b(and)f(ther)-5 b(efor)g(e)28 b Fh(G)g Fi(is)g(eventual)5 b(ly)166 1651 y Fh(k)s Fi(-p)-5 b(erio)g(dic)34 b(of)g(p)-5 b(erio)g(d)35 b Fj(1)f Fi(or)h Fj(2)p Fi(.)99 b Ff(2)166 2118 y Fi(4.2)g(Existenc)-5 b(e)34 b(of)h(R)-5 b(e)g(gular)35 b(Whitney)g(T)-7 b(riangulations)166 2482 y Fj(As)31 b(men)m(tioned)f(b)s(efore,)h(for)f(eac)m(h)h Fh(d)c Fg(2)h(f)p Fj(3)p Fh(;)17 b Fj(4)p Fh(;)g Fj(5)p Fg(g)29 b Fj(there)i(is)f(a)g(unique)h(lo)s(cally)d Fh(C)3112 2497 y Fn(d)3183 2482 y Fj(graph,)166 2602 y(namely)36 b(the)h(tetrahedron,)g(o)s(ctahedron)f(and)h(icosahedron.)f(These)j (three)e(are)f(trian-)166 2723 y(gulations)f(of)h(the)i(sphere)g(\()p Fh(\037)d Fj(=)g(2\).)h(Also,)h(for)f Fh(d)e Fj(=)h(6,)i(w)m(e)h(men)m (tioned)e(that)h(there)g(is)166 2843 y(an)h(in\014nite)e(n)m(um)m(b)s (er)i(of)g(lo)s(cally)d Fh(C)1503 2858 y Ft(6)1579 2843 y Fj(graphs)j(\(see)h([9]\),)f(eac)m(h)g(of)g(them)f(triangulating)166 2963 y(either)32 b(the)h(torus)g(or)g(the)g(Klein)e(b)s(ottle)g(\()p Fh(\037)d Fj(=)f(0\).)166 3207 y(The)34 b(\014rst)f(example)f(of)g(a)h (lo)s(cally)d Fh(C)1522 3222 y Fn(t)1584 3207 y Fj(graph)j(with)f Fh(t)c Fg(\025)h Fj(7)j(that)h(w)m(e)h(studied)f(on)f(a)h(com-)166 3328 y(puter)g(\(using)f(Gap)g([3]\))g(w)m(as)h(a)f(lo)s(cally)e Fh(C)1751 3343 y Ft(10)1858 3328 y Fj(graph)i(whic)m(h)h(is)e(v)m(ery)j (easy)g(to)e(construct)166 3448 y(using)42 b(the)h(pro)s(duct)g Fh(G)29 b Fg(\002)g Fh(H)50 b Fj(with)42 b(\()p Fh(g)t(;)17 b(h)p Fj(\))42 b(adjacen)m(t)h(to)f(\()p Fh(g)2447 3412 y Fe(0)2470 3448 y Fh(;)17 b(h)2570 3412 y Fe(0)2593 3448 y Fj(\))42 b(i\013)g Fg(f)p Fh(g)t(;)17 b(g)2996 3412 y Fe(0)3017 3448 y Fg(g)45 b(2)g Fh(E)6 b Fj(\()p Fh(G)p Fj(\))166 3568 y(and)42 b Fg(f)p Fh(h;)17 b(h)571 3532 y Fe(0)594 3568 y Fg(g)43 b(2)g Fh(E)6 b Fj(\()p Fh(H)i Fj(\).)41 b(W)-8 b(e)42 b(kno)m(w)h(after)f(Prop)s(osition)e(5)h (of)g([6])h(that)f Fh(G)28 b Fg(\002)h Fh(H)49 b Fj(is)41 b(lo-)166 3689 y(cally)31 b Fh(G)473 3704 y Ft(0)535 3689 y Fg(\002)23 b Fh(H)716 3704 y Ft(0)787 3689 y Fj(whenev)m(er)35 b Fh(G)e Fj(is)f(lo)s(cally)e Fh(G)1809 3704 y Ft(0)1881 3689 y Fj(and)j Fh(H)40 b Fj(is)32 b(lo)s(cally)e Fh(H)2680 3704 y Ft(0)2719 3689 y Fj(.)i(Therefore,)i(since)166 3809 y Fh(C)236 3824 y Ft(5)297 3809 y Fg(\002)21 b Fh(K)478 3824 y Ft(2)545 3809 y Fj(=)28 b Fh(C)719 3824 y Ft(10)793 3809 y Fj(,)33 b(the)f(pro)s(duct)g(of)g(the)h(icosahedron)f(and)g(a)g (triangle)e(is)i(a)g(lo)s(cally)d Fh(C)3379 3824 y Ft(10)166 3929 y Fj(graph)j(of)h(order)f(36.)166 4173 y(F)-8 b(or)40 b(eac)m(h)j Fh(d)f Fg(\025)h Fj(7)e(there)h(is)e(an)i(in\014nite)e(n)m (um)m(b)s(er)i(of)e(lo)s(cally)f Fh(C)2600 4188 y Fn(d)2681 4173 y Fj(graphs,)j(as)g(will)c(b)s(e)166 4293 y(sho)m(wn)30 b(in)e(the)h(next)h(prop)s(osition.)c(All)h(this)i(graphs)g (triangulate)d(surfaces)k(of)e(negativ)m(e)166 4414 y(Euler)38 b(c)m(haracteristic)g Fh(\037)f Fj(=)1259 4375 y Fn(n)p 1259 4391 43 4 v 1263 4448 a Ft(6)1312 4414 y Fj(\(6)25 b Fg(\000)i Fh(d)p Fj(\))37 b(and,)i(at)e(v)-5 b(ariance)38 b(with)g(the)g(case)h Fh(d)e Fg(\024)g Fj(6,)h(the)166 4534 y(Euler)i(c)m(haracteristic)f(is)g(a)g(function)g(of)g(the)h(n)m (um)m(b)s(er)g(of)g(v)m(ertices.)g(W)-8 b(e)40 b(will)e(restrict)166 4655 y(to)31 b(orien)m(table)e(surfaces)j(and)f(w)m(e)h(will)d(use)j (triangular)c(co)m(v)m(ering)j(maps)g(and)g(liftings)d(of)166 4775 y(graphs,)33 b(for)f(whic)m(h)h(w)m(e)h(refer)f(to)f([9].)166 5019 y Fk(Prop)s(osition)k(17)48 b Fi(L)-5 b(et)38 b Fh(d)f Fi(b)-5 b(e)37 b(any)g(inte)-5 b(ger)37 b(gr)-5 b(e)g(ater)37 b(than)g(or)g(e)-5 b(qual)37 b(to)g Fj(7)p Fi(.)g(Then)g(ther)-5 b(e)166 5139 y(exists)41 b(an)f(in\014nite)h (numb)-5 b(er)40 b(of)h(non-isomorphic)e(lo)-5 b(c)g(al)5 b(ly)40 b Fh(C)2473 5154 y Fn(d)2555 5139 y Fi(gr)-5 b(aphs.)40 b(F)-7 b(urthermor)i(e,)166 5259 y(ther)g(e)44 b(is)h(an)f(in\014nite)f(numb)-5 b(er)44 b(of)h(non-home)-5 b(omorphic)41 b(orientable)j(close)-5 b(d)43 b(surfac)-5 b(es)166 5380 y(admitting)34 b(a)h(Whitney)g(triangulation)g(which)f (is)g(r)-5 b(e)g(gular)35 b(of)g(de)-5 b(gr)g(e)g(e)34 b Fh(d)p Fi(.)1745 5712 y Fj(12)p eop %%Page: 13 13 13 12 bop 166 83 a Fk(PR)m(OOF.)49 b Fj(M.)34 b(Bro)m(wn)g(and)g(R.)f (Connelly)g(pro)m(v)m(ed)i(in)e([1])g(that)g(there)i(exists)f(at)f (least)166 203 y(one)44 b(\014nite)f(lo)s(cally)d Fh(C)1005 218 y Fn(d)1089 203 y Fj(graph)j Fh(G)p Fj(.)h(If)f(the)h(closed)f (surface)i Fg(S)51 b Fj(asso)s(ciated)43 b(with)g Fh(G)g Fj(is)166 324 y(not)g(orien)m(table,)f(w)m(e)i(can)g(lift)d Fh(G)i Fj(to)g(a)f(graph)2001 291 y Fa(e)1977 324 y Fh(G)h Fj(whose)i(asso)s(ciated)e(surface)h(is)e(the)166 444 y(orien)m(tation)c(co)m(v)m(er)954 412 y Fa(e)929 444 y Fg(S)48 b Fj(of)39 b Fg(S)7 b Fj(.)41 b(In)g(that)f(case,)h(w)m(e)g (ha)m(v)m(e)g(a)f(2-to-1)e(triangular)g(co)m(v)m(ering)166 565 y(map)29 b(of)h(graphs)825 532 y Fa(e)801 565 y Fh(G)e Fg(!)f Fh(G)j Fj(and,)g(since)g(triangular)e(co)m(v)m(ering)i(maps)g (of)g(graphs)g(are)g(lo)s(cal)166 685 y(isomorphisms,)829 653 y Fa(e)806 685 y Fh(G)36 b Fj(is)g(also)f(a)i(\014nite)f(lo)s (cally)e Fh(C)1942 700 y Fn(d)2018 685 y Fj(graph.)j(Th)m(us,)h(w)m(e)f (can)g(assume)g(that)166 805 y Fg(S)47 b Fj(is)38 b(orien)m(table.)g (Being)g(orien)m(table)g(and)g(with)h(negativ)m(e)g(Euler)g(c)m (haracteristic,)f(the)166 926 y(surface)33 b Fg(S)41 b Fj(is)32 b(then)h(a)f(m)m(ultiple)e(torus,)j(sa)m(y)h(a)e(sphere)i (with)e Fh(g)f(>)d Fj(1)k(handles.)166 1147 y(No)m(w,)44 b(for)f(eac)m(h)h(in)m(teger)f Fh(n)k Fg(\025)f Fj(1,)d(there)i(exists) f(an)f(orien)m(table)f(closed)i(surface)g Fg(S)3351 1111 y Ft(\()p Fn(n)p Ft(\))166 1267 y Fj(\(namely)33 b(a)g(sphere)i(with)e Fh(n)p Fj(\()p Fh(g)26 b Fg(\000)d Fj(1\))g(+)f(1)33 b(handles\))h(suc)m(h)h(that)e(there)h(exists)h(an)e Fh(n)p Fj(-to-1)166 1388 y(co)m(v)m(ering)h(map)f Fg(S)834 1351 y Ft(\()p Fn(n)p Ft(\))967 1388 y Fg(!)c(S)7 b Fj(.)35 b(Indeed,)g(there)g(is)e(an)h Fh(n)p Fj(-to-1)e(co)m(v)m(ering)i(map)f Fh(')d Fj(=)g Fh(z)3199 1351 y Fn(n)3269 1388 y Fg(\002)24 b Fh(id)166 1508 y Fj(from)30 b(the)j(2-dimensional)28 b(torus)k Fd(T)f Fj(=)d Fd(S)1682 1466 y Ft(1)1742 1508 y Fg(\002)20 b Fd(S)1901 1472 y Ft(1)1966 1508 y Fj(to)31 b(itself.)g(Remo)m(v)m(e)h(a)f(small)f(op)s(en)h(disk)166 1628 y Fh(D)j Fj(from)c Fd(T)35 b Fj(and)c(use)i(it)d(to)h(form)f(the)i (connected)h(sum)e Fd(T)p Fj(#)p Fg(S)2444 1592 y Fe(0)2503 1628 y Fj(of)g Fd(T)k Fj(and)c(an)h(orien)m(table)166 1749 y(surface)c Fg(S)556 1713 y Fe(0)608 1749 y Fj(of)e(gen)m(us)j Fh(g)15 b Fg(\000)c Fj(1.)28 b(This)f(connected)i(sum)f(is)f (homeomorphic)e(to)i Fg(S)7 b Fj(.)28 b(Consider)166 1869 y(no)m(w)41 b Fh(')441 1833 y Fe(\000)p Ft(1)535 1869 y Fj(\()p Fh(D)s Fj(\))f Fg(\032)h Fd(T)p Fj(,)i(whic)m(h)e(is)e (the)h(disjoin)m(t)f(union)h(of)f Fh(n)h Fj(op)s(en)g(disks)h(b)s (ecause)h Fh(D)g Fj(is)166 1990 y(small.)27 b(Remo)m(ving)i(these)h Fh(n)g Fj(disks)g(from)e Fd(T)33 b Fj(and)c(attac)m(hing)g Fh(n)h Fj(copies)f(of)g Fg(S)2956 1953 y Fe(0)3010 1990 y Fj(\(eac)m(h)h(with)166 2110 y(a)i(small)f(disk)h(remo)m(v)m(ed\))i (w)m(e)f(obtain)f(the)h(surface)g Fg(S)2142 2074 y Ft(\()p Fn(n)p Ft(\))2245 2110 y Fj(.)166 2331 y(F)-8 b(or)37 b(eac)m(h)j Fh(n)d(>)g Fj(1,)h(w)m(e)i(can)e(use)h(the)g(co)m(v)m (ering)g(map)e Fg(S)2253 2295 y Ft(\()p Fn(n)p Ft(\))2393 2331 y Fg(!)g(S)46 b Fj(to)38 b(lift)e(the)j(graph)f Fh(G)166 2451 y Fj(to)33 b(a)g(graph)f Fh(G)721 2415 y Ft(\()p Fn(n)p Ft(\))856 2451 y Fj(whic)m(h)i(underlies)f(a)f (triangulation)e(of)j Fg(S)2395 2415 y Ft(\()p Fn(n)p Ft(\))2497 2451 y Fj(.)h(Since)f(w)m(e)h(then)g(ha)m(v)m(e)g(a)166 2572 y(triangular)23 b(co)m(v)m(ering)j(map)f(of)g(graphs)g Fh(G)1687 2536 y Ft(\()p Fn(n)p Ft(\))1817 2572 y Fg(!)i Fh(G)f Fj(and)f(these)i(are)e(lo)s(cal)f(isomorphisms,)166 2692 y Fh(G)243 2656 y Ft(\()p Fn(n)p Ft(\))377 2692 y Fj(is)32 b(also)g(a)g(\014nite)h(lo)s(cally)d Fh(C)1381 2707 y Fn(d)1453 2692 y Fj(graph.)98 b Ff(2)166 3042 y Fi(4.3)h(Pr)-5 b(o)g(of)35 b(of)f(The)-5 b(or)g(em)34 b(12)166 3384 y Fj(If)24 b Fh(G)g Fj(is)f(lo)s(cally)e(cyclic,)j(the)g (lo)s(cal)e(girth)h(at)g(a)h(giv)m(en)g(v)m(ertex)i(is)d(the)h(degree)h (of)e(the)i(v)m(ertex,)166 3504 y(so)40 b(the)f(lo)s(cal)e(girth)i(of)f Fh(G)i Fj(is)e(its)h(minim)m(um)d(degree)k Fh(\016)t Fj(\()p Fh(G)p Fj(\).)g(Therefore,)g(if)e Fh(\016)t Fj(\()p Fh(G)p Fj(\))h Fg(\025)h Fj(7,)166 3624 y(w)m(e)34 b(already)e(kno)m(w) j(b)m(y)e(Theorem)h(8)e(that)h Fh(k)s(G)g Fj(is)f(clique-Helly)-8 b(.)31 b(Using)i(Theorem)g(7\(2\),)166 3745 y(Theorem)c(12)f(will)e (then)j(b)s(e)f(a)g(consequence)j(of)d(the)h(follo)m(wing)d(result,)i (whic)m(h)h(do)s(es)g(not)166 3865 y(need)34 b(the)f(h)m(yp)s(othesis)g (on)g(the)g(minim)m(um)c(degree:)166 4086 y Fk(Prop)s(osition)36 b(18)48 b Fi(If)25 b Fh(G)g Fi(is)g(lo)-5 b(c)g(al)5 b(ly)24 b(cyclic,)h(then)g(no)f(vertex)h(of)g Fh(k)s(G)g Fi(dominates)f(another)166 4207 y(di\013er)-5 b(ent)34 b(vertex)h(of)g Fh(k)s(G)p Fi(.)166 4557 y Fk(PR)m(OOF.)49 b Fj(The)33 b(only)f(lo)s(cally)d(cyclic)j(graph)g(with)g(minim)m(um)d (degree)k(3)f(is)g(the)g(tetra-)166 4677 y(hedron)k Fh(K)577 4692 y Ft(4)616 4677 y Fj(.)g(Since)f Fh(k)s(K)1073 4692 y Ft(4)1148 4677 y Fj(has)g(just)h(one)f(v)m(ertex,)i(the)f(result)f (is)g(true)g(when)i Fh(\016)t Fj(\()p Fh(G)p Fj(\))32 b(=)g(3.)166 4798 y(W)-8 b(e)48 b(can)f(therefore)h(assume)g(that)f Fh(\016)t Fj(\()p Fh(G)p Fj(\))53 b Fg(\025)g Fj(4,)47 b(and)g(hence)i Fh(G)e Fj(can)h(not)f(con)m(tain)g(a)166 4918 y(tetrahedron.)166 5139 y(The)33 b(cliques)f(of)f Fh(G)g Fj(are)h(clearly)f(its)g(triangles,)g(so)h Fh(k)s(G)f Fj(has)i(a)e(v)m(ertex)j(for)d(eac)m(h)i(triangle)166 5259 y(of)27 b Fh(G)g Fj(and)g(t)m(w)m(o)h(di\013eren)m(t)g(v)m (ertices)g Fh(T)41 b Fj(and)27 b Fh(T)1819 5223 y Fe(0)1869 5259 y Fj(of)g Fh(k)s(G)g Fj(are)g(adjacen)m(t)h(if)e(and)i(only)e(if)g (these)166 5380 y(triangles)31 b(meet)i(at)f(either)g(a)h(v)m(ertex)h (or)e(an)h(edge.)1745 5712 y(13)p eop %%Page: 14 14 14 13 bop 166 83 a Fj(Let)40 b Fh(T)54 b Fg(6)p Fj(=)40 b Fh(T)646 47 y Fe(0)708 83 y Fj(b)s(e)g(t)m(w)m(o)h(triangles)d(of)i Fh(G)p Fj(.)f(W)-8 b(e)41 b(will)c(sho)m(w)k(that)f Fh(T)53 b Fj(do)s(es)40 b(not)g(dominate)166 203 y Fh(T)237 167 y Fe(0)260 203 y Fj(,)g(i.e.,)f(that)g(either)g Fh(T)53 b Fj(and)40 b Fh(T)1397 167 y Fe(0)1459 203 y Fj(are)f(not)h(neigh)m(b) s(ours,)f(or)g(they)i(are)e(but)h(there)g(is)f(a)166 324 y(neigh)m(b)s(our)31 b Fh(T)691 288 y Fe(00)764 324 y Fj(of)g Fh(T)945 288 y Fe(0)999 324 y Fj(whic)m(h)h(is)f(not)g(a)g (neigh)m(b)s(our)g(of)g Fh(T)14 b Fj(.)31 b(Let)g(us)h(assume)g(then)g (that)f Fh(T)166 444 y Fj(and)i Fh(T)427 408 y Fe(0)482 444 y Fj(are)g(neigh)m(b)s(ours)g(in)e Fh(k)s(G)p Fj(.)166 664 y(Assume)47 b(\014rst)f(that)g Fh(T)59 b Fj(and)46 b Fh(T)1370 628 y Fe(0)1439 664 y Fj(meet)g(at)f(an)h(edge,)g(sa)m(y)h Fh(T)64 b Fj(=)50 b Fg(f)p Fh(a;)17 b(b;)g(c)p Fg(g)46 b Fj(and)g Fh(T)3305 628 y Fe(0)3378 664 y Fj(=)166 785 y Fg(f)p Fh(a;)17 b(b;)g(c)438 749 y Fe(0)461 785 y Fg(g)p Fj(.)26 b(Since)g Fh(\016)t Fj(\()p Fh(G)p Fj(\))i Fh(>)f Fj(3,)f(in)f(the)i(cycle)f(of)g(neigh)m(b)s(ours)g(of)f Fh(c)2486 749 y Fe(0)2536 785 y Fj(there)h(is)g(an)g(edge)g Fg(f)p Fh(d;)17 b(e)p Fg(g)166 905 y Fj(whic)m(h)26 b(is)f(disjoin)m(t) f(from)g Fg(f)p Fh(a;)17 b(b)p Fg(g)p Fj(.)26 b(Consider)g(the)g (triangle)d Fh(T)2363 869 y Fe(00)2433 905 y Fj(=)28 b Fg(f)p Fh(c)2629 869 y Fe(0)2652 905 y Fh(;)17 b(d;)g(e)p Fg(g)p Fj(,)24 b(whic)m(h)i(meets)166 1025 y Fh(T)237 989 y Fe(0)289 1025 y Fj(in)i Fh(c)441 989 y Fe(0)464 1025 y Fj(.)h(W)-8 b(e)29 b(claim)d(that)j Fh(T)f Fg(\\)14 b Fh(T)1384 989 y Fe(00)1455 1025 y Fj(=)27 b Fd(?)p Fj(.)j(Indeed,)g(if)e(this)g(w)m(ere)i(not)f(the)g(case)h(w)m(e)g(w)m (ould)166 1146 y(ha)m(v)m(e)k Fh(c)28 b Fg(2)g(f)p Fh(d;)17 b(e)p Fg(g)p Fj(,)32 b(and)g Fh(G)h Fj(w)m(ould)f(con)m(tain)g(a)h (tetrahedron.)166 1366 y(Assume)46 b(no)m(w)f(that)f Fh(T)59 b Fj(and)44 b Fh(T)1366 1330 y Fe(0)1434 1366 y Fj(meet)h(at)f(a)g(v)m(ertex,)j(sa)m(y)e Fh(T)62 b Fj(=)49 b Fg(f)p Fh(a;)17 b(b;)g(c)p Fg(g)44 b Fj(and)h Fh(T)3307 1330 y Fe(0)3378 1366 y Fj(=)166 1486 y Fg(f)p Fh(a;)17 b(b)352 1450 y Fe(0)376 1486 y Fh(;)g(c)462 1450 y Fe(0)485 1486 y Fg(g)p Fj(.)43 b(Since)g(eac)m(h)h(edge)g(of)f Fh(G)g Fj(is)g(con)m(tained)h(in)e(exactly)i(t)m(w)m(o)g(triangles,)e (there)166 1607 y(m)m(ust)48 b(exist)g(a)g(triangle)e Fh(T)1210 1571 y Fe(00)1306 1607 y Fj(=)53 b Fg(f)p Fh(a)1536 1571 y Fe(0)1560 1607 y Fh(;)17 b(c)1646 1571 y Fe(0)1669 1607 y Fh(;)g(b)1754 1571 y Fe(0)1777 1607 y Fg(g)48 b Fj(whic)m(h)g(meets)g Fh(T)2532 1571 y Fe(0)2603 1607 y Fj(in)f(the)i(edge)f Fg(f)p Fh(b)3244 1571 y Fe(0)3268 1607 y Fh(;)17 b(c)3354 1571 y Fe(0)3377 1607 y Fg(g)p Fj(.)166 1727 y(Once)32 b(again)e(w)m(e)i(m)m(ust)f(ha)m(v)m(e)h(that)f Fh(T)i Fg(\\)20 b Fh(T)1735 1691 y Fe(00)1805 1727 y Fj(=)27 b Fd(?)p Fj(,)32 b(b)s(ecause)g(otherwise)g(w)m(e)g(w)m(ould)f (ha)m(v)m(e)166 1847 y Fh(a)217 1811 y Fe(0)268 1847 y Fg(2)d(f)p Fh(b;)17 b(c)p Fg(g)33 b Fj(and)f Fh(G)h Fj(w)m(ould)f(con)m(tain)g(a)h(tetrahedron.)98 b Ff(2)166 2249 y Fk(Ac)m(kno)m(wledgemen)m(ts)166 2589 y Fj(W)-8 b(e)32 b(w)m(ould)g(lik)m(e)f(to)g(express)j(our)e(thanks)h(to)e(the)h (referee\(s\))h(for)f(the)g(v)-5 b(aluable)30 b(sugges-)166 2710 y(tions)i(that)g(impro)m(v)m(ed)h(this)f(presen)m(tation.)166 3097 y Fk(References)166 3430 y Fq([1])71 b(M.)22 b(Bro)m(wn)f(and)g (R.)g(Connelly)-8 b(.)20 b Fs(On)k(Gr)-5 b(aphs)26 b(With)f(a)f (Constant)i(Link)p Fq(,)21 b(in)f(New)h(Directions)332 3543 y(in)i(the)g(Theory)h(of)f(Graphs)g(\(Pro)s(c.)h(Third)d(Ann)i (Arb)s(or)f(Conf.)h(on)h(Graph)f(Theory)-8 b(,)24 b(Univ.)332 3656 y(of)31 b(Mic)m(higan,)f(Ann)g(Arb)s(or,)f(Mic)m(h.,)i(1971\))h (Academic)f(Press,)f(N.Y.)h(\(1973\))i(19-51.)166 3842 y([2])71 b(F.)27 b(Escalan)m(te.)889 3819 y Fs(\177)873 3842 y(Ub)-5 b(er)29 b(Iterierte)g(Clique-Gr)-5 b(aphen)p Fq(.)28 b(Abh.)e(Math.)i(Sem.)e(Univ.)g(Ham)m(burg.)332 3955 y(39)32 b(\(1973\))g(58-68.)166 4142 y([3])71 b(The)55 b(GAP)g(Group,)f(Aac)m(hen,)i(St)f(Andrews.)f Fs(GAP)g({)i(Gr)-5 b(oups,)57 b(A)n(lgorithms,)f(and)332 4255 y(Pr)-5 b(o)g(gr)g(amming,) 35 b(V)-7 b(ersion)33 b(4.2)p Fq(,)f(2000.)g(\(h)m (ttp://www-gap.dcs.st-and.ac.uk/)h(gap\).)166 4442 y([4])71 b(N.)38 b(Harts\014eld)f(and)g(G.)i(Ringel.)d Fs(Cle)-5 b(an)41 b(T)-7 b(riangulations.)39 b Fq(Com)m(binatorica)e(11)i (\(1991\),)332 4555 y(145-155.)166 4741 y([5])71 b(S.)43 b(T.)g(Hedetniemi)f(and)g(P)-8 b(.)44 b(J.)f(Slater.)f Fs(Line)i(Gr)-5 b(aphs)47 b(of)e(T)-7 b(riangleless)45 b(Gr)-5 b(aphs)47 b(and)332 4854 y(Iter)-5 b(ate)g(d)31 b(Clique)c(Gr)-5 b(aphs.)28 b Fq(Springer)c(Lecture)i(Notes)g(in)f (Mathematics)h(303)h(\(1972\))h(139-)332 4967 y(147.)166 5154 y([6])71 b(P)-8 b(.)42 b(Hell.)f Fs(Gr)-5 b(aphs)46 b(with)e(Given)e(Neighb)-5 b(orho)g(o)g(ds)47 b(I)p Fq(,)41 b(in)g(Probl)m(\022)-43 b(emes)41 b(Com)m(binatoires)g(et)332 5267 y(Th)m(\023)-43 b(eorie)26 b(des)f(Graphes)h(\(Collo)s(ques)e(in)m (ternationaux)g(CNRS,)h(260\).)j(P)m(aris,)d(Editions)f(du)332 5380 y(CNRS)30 b(\(1978\),)j(219-223.)1745 5712 y Fj(14)p eop %%Page: 15 15 15 14 bop 166 83 a Fq([7])71 b(L.)52 b(C.)f(Kinsey)-8 b(.)50 b Fs(T)-7 b(op)i(olo)g(gy)55 b(of)d(Surfac)-5 b(es)p Fq(.)52 b(Undergraduate)f(T)-8 b(exts)52 b(in)e(Mathematics.)332 196 y(Springer-V)-8 b(erlag,)30 b(N.Y.)h(\(1993\).)166 382 y([8])71 b(F.)42 b(Larri\023)-45 b(on)40 b(and)h(V.)h (Neumann-Lara.)f Fs(Clique)h(Diver)-5 b(gent)43 b(Gr)-5 b(aphs)45 b(with)e(Unb)-5 b(ounde)g(d)332 495 y(Se)g(quenc)g(e)33 b(of)g(Diameters)p Fq(.)e(Discrete)g(Mathematics)g(197-198)j(\(1999\))e (491-501.)166 682 y([9])71 b(F.)45 b(Larri\023)-45 b(on)42 b(and)h(V.)h(Neumann-Lara.)f Fs(L)-5 b(o)g(c)g(al)5 b(ly)55 b Fp(C)2180 696 y Ft(6)2264 682 y Fs(Gr)-5 b(aphs)48 b(ar)-5 b(e)45 b(Clique)g(Diver)-5 b(gent)p Fq(.)332 795 y(Discrete)31 b(Mathematics)h(215)f(\(2000\))i(159-170.)166 981 y([10])26 b(V.)g(Neumann-Lara.)f Fs(On)j(Clique-diver)-5 b(gent)28 b(Gr)-5 b(aphs)p Fq(,)28 b(in)23 b(Probl)m(\022)-43 b(emes)26 b(Com)m(binatoires)e(et)332 1094 y(Th)m(\023)-43 b(eorie)26 b(des)f(Graphes)h(\(Collo)s(ques)e(in)m(ternationaux)g (CNRS,)h(260\).)j(P)m(aris,)d(Editions)f(du)332 1207 y(CNRS)30 b(\(1978\),)j(313-315.)166 1393 y([11])26 b(M.)31 b(A.)g(Piza)s(~)-48 b(na.)30 b Fs(The)j(Ic)-5 b(osahe)g(dr)g(on)37 b(is)32 b(Clique-Diver)-5 b(gent)p Fq(.)30 b(Submitted.)166 1579 y([12])c(E.)42 b(Prisner.)e Fs(Gr)-5 b(aph)45 b(Dynamics)p Fq(.)d(Pitman)f(Researc)m(h)h(Notes)h(in)d(Mathematics)j(Series)332 1692 y(338.)32 b(Longman,)f(Essex)f(\(1995\).)166 1879 y([13])c(J.)d(L.)f(Szw)m(arc\014ter.)h Fs(A)i(Survey)g(on)h(Clique)e (Gr)-5 b(aphs)p Fq(,)26 b(in)21 b(Recen)m(t)i(Adv)-5 b(ances)23 b(in)e(Algorithms)332 1992 y(and)26 b(Com)m(binatorics.)g (C.)h(Linhares)e(and)h(B.)h(Reed,)g(eds.,)g(Springer-V)-8 b(erlag.)26 b(T)-8 b(o)27 b(app)s(ear.)166 2178 y([14])f(W.)44 b(T.)g(T)-8 b(utte.)44 b Fs(A)g(c)-5 b(ensus)45 b(of)f(plane)i (triangulations.)f Fq(Canad.)e(J.)h(Math.)g(14)g(\(1962\),)332 2291 y(21-28.)166 2477 y([15])26 b(H.)31 b(Whitney)-8 b(.)31 b Fs(A)h(The)-5 b(or)g(em)34 b(on)f(Gr)-5 b(aphs)p Fq(.)33 b(Ann.)d(Math.)h(\(2\))h(32)f(\(1931\))i(378-390.)1745 5712 y Fj(15)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF
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