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b(Cohen)h(and)f (Neumaier)g([6],)i(Chapter)e(23)h(of)g(Biggs)g([2],)h(and)e(Chapter)g (6)i(of)f(Holton)60 1005 y(and)j(Sheehan)g([8].)23 b(F)l(or)16 b(a)h(surv)o(ey)f(of)i(results)e(on)g(cubic)h(cages)g(\()p Fd(k)f Fl(=)e(3\))k(with)f(girth)f(at)i(most)60 1065 y(20,)e(see)h(Ro)o(yle)f([18].)160 1187 y(Finding)21 b(upp)q(er)h(b)q(ounds)f(for)i Fd(v)r Fl(\()p Fd(k)r(;)8 b(g)r Fl(\))23 b(is)f(a)h(far)f(more)g(di\016cult)g(a\013air;)k (indeed,)d(ev)o(en)60 1247 y(the)18 b(fact)h(that)g Fd(v)r Fl(\()p Fd(k)r(;)8 b(g)r Fl(\))18 b(is)g(\014nite)g(is)g(non)o(trivial) e(to)j(pro)o(v)o(e.)25 b(This)18 b(w)o(as)f(settled)i(b)o(y)f(Sac)o(hs) f([19])60 1307 y(who)g(sho)o(w)o(ed)e(b)o(y)i(explicit)h(construction)e (that)i(\()p Fd(k)r(;)8 b(g)r Fl(\)-graphs)15 b(of)i(\014nite)g(order)f (exist.)25 b(In)17 b(the)60 1366 y(same)e(y)o(ear,)g(Erd})-25 b(os)15 b(and)g(Sac)o(hs)g([7])g(ga)o(v)o(e,)g(without)h(explicit)g (construction,)f(a)h(m)o(uc)o(h)e(smaller)60 1426 y(general)i(upp)q(er) g(b)q(ound)h(on)f Fd(v)r Fl(\()p Fd(k)r(;)8 b(g)r Fl(\).)24 b(\(As)17 b(w)o(as)f(p)q(oin)o(ted)h(b)o(y)f(Alon)h(in)g([1,)g(p.)f (1752],)h(although)60 1486 y(their)e(pro)q(of)h(do)q(es)f(supply)g(a)g (p)q(olynomial)g(time)h(algorithm)e(for)h(constructing)g(graphs)f(whic) o(h)60 1546 y(pro)o(vide)h(the)i(upp)q(er)f(b)q(ound,)f(their)h(graphs) f(are)h(not)h(really)f(explicit)g(in)g(the)h(sense)f(that)h(it)g(is)60 1605 y(not)g(clear)f(ho)o(w)g(to)h(decide)f(e\016cien)o(tly)h(whether)f (or)g(not)h(t)o(w)o(o)f(prescrib)q(ed)f(v)o(ertices)i(of)g(suc)o(h)e(a) 60 1665 y(graph)i(are)h(adjacen)o(t.\))27 b(Their)17 b(result)h(w)o(as)f(later)h(impro)o(v)o(ed,)f(though)g(sligh)o(tly)l(,) h(b)o(y)g(W)l(alther)60 1725 y([21],)e([22])g(and)g(b)o(y)g(Sauer)g ([20].)21 b(The)16 b(follo)o(wing)g(upp)q(er)f(b)q(ounds)h(are)g(due)g (to)h(Sauer)e([20]:)401 1913 y Fd(v)r Fl(\()p Fd(k)r(;)8 b(g)r Fl(\))13 b Fc(\024)607 1843 y Fa(\032)653 1885 y Fl(2\()p Fd(k)g Fc(\000)d Fl(1\))829 1867 y Fi(g)q Fh(\000)p Fm(2)954 1885 y Fl(for)16 b Fd(g)i Fl(o)q(dd)e(and)g Fd(k)f Fc(\025)f Fl(4,)i(and)653 1944 y(4\()p Fd(k)d Fc(\000)d Fl(1\))829 1926 y Fi(g)q Fh(\000)p Fm(3)954 1944 y Fl(for)16 b Fd(g)i Fl(ev)o(en)e(and)g Fd(k)g Fc(\025)d Fl(4.)1796 1913 y(\(1\))60 2098 y(Note)19 b(that)g(these)f(upp)q(er)g (b)q(ounds)f(are)h(roughly)f(the)h(squares)f(of)i(the)g(previously)e (indicated)60 2158 y(lo)o(w)o(er)e(b)q(ounds.)160 2280 y(In)i(this)h(pap)q(er)g(w)o(e)f(establish)g(general)h(upp)q(er)f(b)q (ounds)g(on)g Fd(v)r Fl(\()p Fd(k)r(;)8 b(g)r Fl(\))18 b(whic)o(h)f(are)h(roughly)60 2340 y(the)g(3/2)g(p)q(o)o(w)o(er)e(of)j (the)f(lo)o(w)o(er)e(b)q(ounds,)h(and)g(w)o(e)h(pro)o(vide)e(explicit)i (constructions)f(for)g(suc)o(h)60 2400 y(\()p Fd(k)r(;)8 b(g)r Fl(\)-graphs.)20 b(The)c(main)g(results)f(are)h(describ)q(ed)g(b) q(elo)o(w.)948 2520 y(2)p eop 3 2 bop 60 -70 a Fb(the)16 b(electr)o(onic)i(journal)f(of)f(combina)m (torics)h(4)f(\(no.)21 b(2\))16 b(\(1997\))g(,)g(#R13)378 b(3)60 50 y Fn(Theorem)20 b(A.)f Fk(L)m(et)i Fd(k)e Fc(\025)f Fl(2)h Fk(and)i Fd(g)f Fc(\025)d Fl(5)j Fk(b)m(e)h(inte)m(gers,)g(and)g (let)e Fd(q)j Fk(denote)f(the)f(smal)s(lest)f(o)m(dd)60 110 y(prime)f(p)m(ower)h(for)g(which)g Fd(k)c Fc(\024)f Fd(q)r Fk(.)23 b(Then)759 260 y Fd(v)r Fl(\()p Fd(k)r(;)8 b(g)r Fl(\))14 b Fc(\024)g Fl(2)p Fd(k)r(q)1049 226 y Ff(3)p 1048 232 17 2 v 1048 251 a(4)1071 239 y Fi(g)q Fh(\000)p Fi(a)1147 260 y Fd(;)635 b Fl(\(2\))60 409 y Fk(wher)m(e)19 b Fd(a)14 b Fl(=)f(4)p Fk(,)18 b Fl(11)p Fd(=)p Fl(4)p Fk(,)g Fl(7)p Fd(=)p Fl(2)p Fk(,)g Fl(13)p Fd(=)p Fl(4)f Fk(for)i Fd(g)c Fc(\021)f Fl(0)p Fd(;)8 b Fl(1)p Fd(;)g Fl(2)p Fd(;)g Fl(3)14 b(mo)q(d)f(4)p Fk(,)18 b(r)m(esp)m(e)m(ctively.)160 579 y Fl(The)11 b(upp)q(er)f(b)q(ounds)g(in)h(\(2\))h(are)f(b)q(etter)h(the)g(ones)e (pro)o(vided)g(in)h(\(1\))h(for)f(all)g Fd(k)k Fc(\025)f Fl(5)d(and)g Fd(g)k Fc(\025)60 639 y Fl(5.)60 699 y(F)l(or)20 b Fd(k)i Fc(\025)f Fd(t)f Fc(\025)h Fl(3)g(and)f Fd(q)j Fc(\021)d Fl(1)h(mo)q(d)13 b Fd(t)p Fl(,)22 b(\()p Fd(k)r(;)8 b Fl(2)p Fd(t)p Fl(\)-graphs)19 b(of)i(orders)e(at)i(least)g(as)f (large)g(as)h(the)60 759 y(upp)q(er)h(b)q(ound)h(in)f(\(2\))i(w)o(ere)f (constructed)f(in)h([9])g(b)o(y)g(F)q(\177)-26 b(uredi,)23 b(Seress,)h(and)e(the)h(authors.)60 818 y(The)c(fact)g(that)h(the)f (orders)e(of)j(these)f(constructions)e(actually)i(meet)g(the)g(upp)q (er)f(b)q(ound)g(in)60 878 y(\(2\))i(for)f Fd(q)j Fl(o)q(dd)d(follo)o (ws)f(from)h([13].)30 b(The)19 b(constructions)f(w)o(e)h(in)o(tro)q (duce)g(in)g(this)g(pap)q(er)f(are)60 938 y(indep)q(enden)o(t)d(of)i (the)g(relativ)o(e)f(magnitudes)f(of)i Fd(k)h Fl(and)e Fd(g)r Fl(.)60 1108 y Fn(Constructions.)25 b Fk(F)l(or)18 b(al)s(l)h Fd(k)d Fc(\025)g Fl(3)i Fk(and)i Fd(g)d Fc(\025)e Fl(6)p Fk(,)k Fd(g)h Fk(even,)f(we)g(explicitly)f(c)m(onstruct)h(a)g Fl(\()p Fd(k)r(;)8 b(g)r Fl(\))p Fk(-)60 1168 y(gr)m(aph)20 b(of)e(or)m(der)650 1270 y Fd(g)683 1215 y Fa(\020)713 1270 y Fl(1)11 b(+)g(\()p Fd(k)i Fc(\000)e Fl(2\))p Fd(k)r(q)1003 1249 y Fi(g)q Fh(\000)p Fm(5)p Fh(\000b)1130 1233 y Fg(g)q Fe(\000)p Ff(3)p 1130 1242 63 2 v 1152 1261 a(4)1198 1249 y Fh(c)1219 1215 y Fa(\021)1257 1270 y Fl(;)525 b(\(3\))60 1410 y Fk(for)19 b(al)s(l)e Fd(k)e Fc(\025)f Fl(3)j Fk(and)i Fd(g)d Fc(\025)d Fl(5)p Fk(,)18 b Fd(g)i Fk(o)m(dd,)g(we)d(explicitly)h(c)m(onstruct)g(a)g Fl(\()p Fd(k)r(;)8 b(g)r Fl(\))p Fk(-gr)m(aph)20 b(of)e(or)m(der)587 1575 y Fl(\()p Fd(g)13 b Fl(+)e(1\))745 1520 y Fa(\020)775 1575 y Fl(1)g(+)g(\()p Fd(k)i Fc(\000)e Fl(2\))p Fd(k)r(q)1065 1554 y Fi(g)q Fh(\000)p Fm(4)p Fh(\000b)1192 1538 y Fg(g)q Fe(\000)p Ff(2)p 1192 1547 V 1214 1566 a(4)1260 1554 y Fh(c)1281 1520 y Fa(\021)1319 1575 y Fd(:)463 b Fl(\(4\))60 1737 y Fk(In)18 b(either)g(c)m(ase,)h Fd(q)h Fk(is)e(the)g(smal)s(lest) f(o)m(dd)j(prime)e(p)m(ower)i(for)e(which)h Fd(k)c Fc(\024)f Fd(q)r Fk(.)160 1857 y Fl(Though)j(the)j(upp)q(er)d(b)q(ounds)h(on)h Fd(v)r Fl(\()p Fd(k)r(;)8 b(g)r Fl(\))18 b(pro)o(vided)g(b)o(y)g(these) h(constructions)f(are)g(not)60 1916 y(as)h(go)q(o)q(d)h(as)g(the)g (ones)f(giv)o(en)h(in)f(Theorem)g(A,)h(they)g(are)f(b)q(etter)i(than)e (the)h(b)q(ounds)f(in)g(\(1\))60 1976 y(for)h(all)g Fd(k)i Fc(\025)e Fl(7)g(and)g Fd(g)h Fc(\025)f Fl(11)g(when)g Fd(g)i Fl(is)e(o)q(dd,)h(and)f Fd(g)h Fc(\025)f Fl(8)g(when)g Fd(g)i Fl(is)e(ev)o(en.)33 b(With)21 b(some)60 2036 y(additional)13 b(restrictions)g(on)g Fd(k)j Fl(and)d Fd(g)j Fl(these)e(constructiv)o (e)g(upp)q(er)f(b)q(ounds)g(can)h(b)q(e)g(impro)o(v)o(ed)60 2096 y(still)i(further.)160 2215 y(By)21 b(Cheb)o(yshev's)f(Theorem,)h (for)g(a)g(\014xed)g(in)o(teger)g Fd(k)i Fc(\025)e Fl(3)g(there)g(is)g (alw)o(a)o(ys)f(a)h(prime)60 2275 y(b)q(et)o(w)o(een)c Fd(k)i Fl(and)d(2)p Fd(k)d Fc(\000)f Fl(2.)24 b(F)l(or)16 b(an)o(y)h Fd(\017)e(>)g Fl(0)i(and)f Fd(k)h Fc(\025)e Fd(k)1109 2282 y Fm(0)1131 2275 y Fl(\()p Fd(\017)p Fl(\),)j(this)f(in) o(terv)m(al)g(can)g(b)q(e)g(narro)o(w)o(ed)60 2340 y(to)g([)p Fd(k)r(;)8 b(k)k Fl(+)f Fd(k)307 2309 y Ff(3)p 306 2315 17 2 v 306 2334 a(5)329 2322 y Fm(+)p Fi(\017)379 2340 y Fl(],)16 b(see)h([17,)f(p.)h(131].)k(Th)o(us)16 b(the)h(upp)q(er)e(b) q(ounds)h(from)g(\(2\),)h(\(3\),)g(and)f(\(4\))h(are)60 2400 y(roughly)e(the)i(3/2)g(p)q(o)o(w)o(er)e(of)i(the)f(indicated)g (lo)o(w)o(er)g(b)q(ounds.)948 2520 y(3)p eop 4 3 bop 60 -70 a Fb(the)16 b(electr)o(onic)i(journal)f(of)f(combina)m (torics)h(4)f(\(no.)21 b(2\))16 b(\(1997\))g(,)g(#R13)378 b(4)60 50 y Fn(2.)25 b(Preliminary)20 b(Results)g(and)e(Pro)r(of)f(of)i (Theorem)e(A)160 160 y Fl(The)e(p)q(ossibilit)o(y)f(of)h(impro)o(ving)e (the)j(upp)q(er)e(b)q(ounds)g(in)h(\(1\))h(for)f(\()p Fd(k)r(;)8 b(g)r Fl(\))14 b Fc(2)g Fd(A)9 b Fc(\002)f Fd(B)r Fl(,)16 b(where)60 220 y(b)q(oth)d Fd(A)h Fl(and)e Fd(B)k Fl(are)c(certain)h(in\014nite)f(subsets)g(of)i(p)q(ositiv)o(e)f (in)o(tegers,)f(b)q(ecame)h(apparen)o(t)f(after)60 280 y(the)j(disco)o(v)o(eries)e(of)j(certain)e(sp)q(ecial)h(in\014nite)f (families)h(of)g(graphs)e(whic)o(h)h(w)o(e)h(describ)q(e)f(b)q(elo)o(w) 60 340 y(\(see)j(F1,)f(F2)g(and)g(F3\).)160 450 y(F)l(or)j(a)h(\014xed) g(in)o(teger)g Fd(k)h Fc(\025)f Fl(3,)h(let)g Fc(f)p Fd(G)897 457 y Fi(i)914 450 y Fc(g)939 457 y Fi(i)p Fh(\025)p Fm(1)1026 450 y Fl(b)q(e)g(a)f(family)g(of)h Fd(k)r Fl(-regular)d (graphs)g(of)j(in-)60 510 y(creasing)15 b(order)g Fd(v)405 517 y Fi(i)422 510 y Fl(.)22 b(Let)17 b Fd(g)571 517 y Fi(i)603 510 y Fl(denote)g(the)f(girth)g(of)g Fd(G)1066 517 y Fi(i)1083 510 y Fl(.)22 b(The)16 b(family)g Fc(f)p Fd(G)1437 517 y Fi(i)1453 510 y Fc(g)h Fl(is)e(called)h(a)g Fk(family)60 570 y(of)j(gr)m(aphs)g(with)g(lar)m(ge)f(girth)f Fl(if)779 632 y Fd(g)803 639 y Fi(i)833 632 y Fc(\025)c Fd(\015)e Fl(log)986 644 y Fi(k)q Fh(\000)p Fm(1)1062 632 y Fl(\()p Fd(v)1105 639 y Fi(i)1122 632 y Fl(\))60 729 y(for)16 b(some)h(p)q(ositiv)o(e)f(constan)o(t)g Fd(\015)k Fl(and)c(all)g Fd(i)f Fc(\025)f Fl(1.)22 b(The)17 b(lo)o(w)o(er)e(b)q(ound)h(for)h Fd(v)r Fl(\()p Fd(k)r(;)8 b(g)r Fl(\))16 b(sho)o(ws)g(that)60 789 y Fd(\015)g Fc(\024)e Fl(2,)i(but)h(no)f(in\014nite)g(family)g(has)g(b)q(een)g(found)g(for)g (whic)o(h)g Fd(\015)g Fl(=)e(2.)60 900 y Fn(F1.)22 b Fl(Margulis)14 b([16])i(and,)g(indep)q(enden)o(tly)l(,)f(Lub)q(otzky)l (,)i(Phillips)d(and)i(Sarnak)g([15])f(came)i(up)60 959 y(with)k(similar)f(examples)g(of)i(graphs)d(with)i Fd(\015)k Fc(\025)c Fl(4)p Fd(=)p Fl(3)g(and)g(arbitrary)f(large)g(v)m(alency)i (\(they)60 1019 y(turned)e(out)i(to)f(b)q(e)h(so{called)e(Raman)o(ujan) f(graphs\).)35 b(These)21 b(are)g(Ca)o(yley)h(graphs)d(of)j(the)60 1079 y(group)13 b Fd(P)o(G)-8 b(L)295 1086 y Fm(2)317 1079 y Fl(\()p Fd(Z)370 1086 y Fi(q)392 1079 y Fl(\))15 b(with)f(resp)q(ect)g(to)h(a)f(set)g(of)h Fd(p)6 b Fl(+)g(1)14 b(generators,)g(where)f Fd(p)i Fl(and)e Fd(q)k Fl(are)c(distinct)60 1139 y(primes,)j(eac)o(h)h(congruen)o(t)f(to)h(1)h(mo)q(d)f(4,)g(with)g (the)h(Legendre)e(sym)o(b)q(ol)1441 1098 y Fa(\000)1464 1117 y Fi(p)p 1464 1127 21 2 v 1465 1156 a(q)1484 1098 y Fa(\001)1522 1139 y Fl(=)f Fc(\000)p Fl(1.)24 b(Denoted)60 1208 y(b)o(y)16 b Fd(X)174 1190 y Fi(p;q)229 1208 y Fl(,)h(they)g(are)f (\()p Fd(p)c Fl(+)f(1\)-regular)k(bipartite)h(graphs)f(of)i(order)e Fd(q)r Fl(\()p Fd(q)1411 1190 y Fm(2)1445 1208 y Fc(\000)c Fl(1\).)23 b(Margulis)15 b([16])60 1268 y(and,)20 b(indep)q(enden)o (tly)l(,)f(Biggs)g(and)g(Boshier)g([3])h(sho)o(w)o(ed)e(that)i(the)g (asymptotic)g(v)m(alue)f(of)h Fd(\015)60 1328 y Fl(for)14 b(the)h(graphs)e Fd(X)422 1310 y Fi(p;q)491 1328 y Fl(is)h(exactly)i(4) p Fd(=)p Fl(3.)21 b(Moreo)o(v)o(er,)13 b(in)h(b)q(oth)g(pap)q(ers)g(an) g(explicit)h(form)o(ula)d(for)60 1388 y(the)j(girth)f Fd(g)r Fl(\()p Fd(X)354 1369 y Fi(p;q)408 1388 y Fl(\))i(of)e Fd(X)542 1369 y Fi(p;q)612 1388 y Fl(w)o(as)g(found.)20 b(T)l(o)15 b(state)g(their)f(results)g(\(form)o(ulae)f(\(5\),)j(b)q (elo)o(w\))e(w)o(e)60 1447 y(\014rst)i(need)g(the)h(follo)o(wing)e (de\014nition.)160 1558 y(Call)21 b(an)g(in)o(teger)g Fk(go)m(o)m(d)i Fl(if)f(it)g(is)f(not)g(of)h(the)f(form)g(4)1183 1540 y Fi(\013)1211 1558 y Fl(\(8)p Fd(\014)d Fl(+)c(7\))22 b(for)f(an)o(y)g(nonnegativ)o(e)60 1618 y(in)o(tegers)d Fd(\013;)8 b(\014)s Fl(.)28 b(By)19 b(a)g(theorem)f(of)h(Legendre,)f (go)q(o)q(d)h(n)o(um)o(b)q(ers)d(are)j(precisely)f(those)g(whic)o(h)60 1677 y(are)e(represen)o(table)f(as)h(sums)f(of)i(three)f(squares.)21 b(Then)320 1870 y Fd(g)r Fl(\()p Fd(X)410 1850 y Fi(p;q)464 1870 y Fl(\))15 b(=)550 1800 y Fa(\032)596 1841 y Fl(2)p Fc(d)p Fl(2)8 b(log)741 1853 y Fi(p)772 1841 y Fd(q)r Fc(e)231 b Fl(if)17 b Fd(p)1120 1822 y Fh(d)p Fm(2)7 b(log)1216 1831 y Fg(p)1244 1822 y Fi(q)q Fh(e)1296 1841 y Fc(\000)j Fd(q)1369 1823 y Fm(2)1408 1841 y Fl(is)16 b(go)q(o)q(d,)596 1901 y(2)p Fc(d)p Fl(2)8 b(log)741 1913 y Fi(p)772 1901 y Fd(q)13 b Fl(+)e(log)921 1913 y Fi(p)952 1901 y Fl(2)p Fc(e)50 b Fl(otherwise.)1796 1870 y(\(5\))60 2050 y Fn(F2.)39 b Fl(In)21 b([10],)i(Lazebnik)f(and)g (Ustimenk)o(o)f(constructed)h(the)g(family)g(of)g(graphs)f Fd(D)q Fl(\()p Fd(n;)8 b(q)r Fl(\),)60 2110 y Fd(n)14 b Fc(\025)f Fl(2,)k Fd(q)h Fl(a)f(prime)e(p)q(o)o(w)o(er,)g(for)h(whic) o(h)g Fd(\015)g Fc(\025)e Fl(log)972 2122 y Fi(q)994 2110 y Fl(\()p Fd(q)f Fc(\000)e Fl(1\).)22 b(Graphs)15 b Fd(D)q Fl(\()p Fd(n;)8 b(q)r Fl(\))18 b(are)e Fd(q)r Fl(-regular)f(of)60 2170 y(order)i(2)p Fd(q)241 2152 y Fi(n)287 2170 y Fl(and)g(girth)h(at)h(least)f Fd(n)12 b Fl(+)g(4)19 b(\(resp)q(ectiv)o(ely)l(,)g Fd(n)12 b Fl(+)g(5\))18 b(for)h Fd(n)f Fl(ev)o(en)g(\(resp)q(ectiv)o(ely)l(,)h Fd(n)60 2230 y Fl(o)q(dd\).)j(These)16 b(are)g(de\014ned)g(as)g(follo)o (ws.)160 2340 y(Let)e Fd(q)i Fl(b)q(e)e(a)g(prime)f(p)q(o)o(w)o(er,)g (and)g(let)i Fd(P)20 b Fl(and)14 b Fd(L)g Fl(b)q(e)g(t)o(w)o(o)f (copies)g(of)i(the)f(coun)o(tably)f(in\014nite)60 2400 y(dimensional)g(v)o(ector)j(space)f(o)o(v)o(er)g Fd(GF)7 b Fl(\()p Fd(q)r Fl(\).)23 b(In)15 b(order)g(to)h(distinguish)d(b)q(et) o(w)o(een)i(v)o(ectors)h(from)948 2520 y(4)p eop 5 4 bop 60 -70 a Fb(the)16 b(electr)o(onic)i(journal)f(of)f(combina)m (torics)h(4)f(\(no.)21 b(2\))16 b(\(1997\))g(,)g(#R13)378 b(5)60 50 y Fd(P)24 b Fl(and)15 b Fd(L)i Fl(w)o(e)f(use)g(paren)o (theses)f(and)h(brac)o(k)o(ets:)21 b Fd(x)14 b Fc(2)g Fd(P)24 b Fl(will)16 b(b)q(e)h(written)f(as)g(\()p Fd(x)p Fl(\),)i(and)e Fd(y)g Fc(2)e Fd(L)60 110 y Fl(as)h([)p Fd(y)r Fl(].)22 b(Adopting)15 b(the)g(notation)h(for)f(co)q(ordinates)f (of)i(p)q(oin)o(ts)f(and)g(lines)f(in)o(tro)q(duced)h(in)g([10],)60 169 y(namely)l(,)274 266 y(\()p Fd(p)p Fl(\))g(=)e(\()p Fd(p)448 273 y Fm(1)471 266 y Fd(;)8 b(p)518 273 y Fm(11)561 266 y Fd(;)g(p)608 273 y Fm(12)650 266 y Fd(;)g(p)697 273 y Fm(21)739 266 y Fd(;)g(p)786 273 y Fm(22)829 266 y Fd(;)g(p)876 246 y Fh(0)876 279 y Fm(22)918 266 y Fd(;)g(p)965 273 y Fm(23)1008 266 y Fd(;)g(:)g(:)g(:)g(;)g(p)1143 273 y Fi(ii)1174 266 y Fd(;)g(p)1221 246 y Fh(0)1221 279 y Fi(ii)1252 266 y Fd(;)g(p)1299 273 y Fi(i;i)p Fm(+1)1392 266 y Fd(;)g(p)1439 273 y Fi(i)p Fm(+1)p Fi(;i)1532 266 y Fd(;)g(:)g(:)g(:)q Fl(\))p Fd(;)346 363 y Fl([)p Fd(l)q Fl(])13 b(=)h([)p Fd(l)485 370 y Fm(1)507 363 y Fd(;)8 b(l)544 370 y Fm(11)586 363 y Fd(;)g(l)623 370 y Fm(12)665 363 y Fd(;)g(l)702 370 y Fm(21)745 363 y Fd(;)g(l)782 370 y Fm(22)824 363 y Fd(;)g(l)862 343 y Fh(0)861 375 y Fm(22)903 363 y Fd(;)g(l)940 370 y Fm(23)982 363 y Fd(;)g(:)g(:)g(:)h(;)f(l)1108 370 y Fi(ii)1138 363 y Fd(;)g(l)1176 343 y Fh(0)1175 375 y Fi(ii)1206 363 y Fd(;)g(l)1243 370 y Fi(i;i)p Fm(+1)1336 363 y Fd(;)g(l)1373 370 y Fi(i)p Fm(+1)p Fi(;i)1466 363 y Fd(;)g(:)g(:)g(:)p Fl(])p Fd(;)60 448 y Fl(w)o(e)16 b(de\014ne)f(an)h(in\014nite)f (bipartite)g(graph)g Fd(D)q Fl(\()p Fd(q)r Fl(\))j(with)e(the)g(v)o (ertex)g(set)g Fd(P)h Fc([)10 b Fd(L)16 b Fl(as)g(follo)o(ws.)21 b(W)l(e)60 507 y(sa)o(y)16 b(\()p Fd(p)p Fl(\))h(is)g(adjacen)o(t)f(to) h([)p Fd(l)q Fl(])f(if)g(the)h(follo)o(wing)f(relations)f(on)h(their)g (co)q(ordinates)g(hold:)737 604 y Fd(l)752 611 y Fm(11)805 604 y Fc(\000)11 b Fd(p)880 611 y Fm(11)936 604 y Fl(=)j Fd(l)1004 611 y Fm(1)1026 604 y Fd(p)1051 611 y Fm(1)737 679 y Fd(l)752 686 y Fm(12)805 679 y Fc(\000)d Fd(p)880 686 y Fm(12)936 679 y Fl(=)j Fd(l)1004 686 y Fm(11)1046 679 y Fd(p)1071 686 y Fm(1)737 754 y Fd(l)752 761 y Fm(21)805 754 y Fc(\000)d Fd(p)880 761 y Fm(21)936 754 y Fl(=)j Fd(l)1004 761 y Fm(1)1026 754 y Fd(p)1051 761 y Fm(11)737 829 y Fd(l)752 836 y Fi(ii)794 829 y Fc(\000)c Fd(p)868 836 y Fi(ii)913 829 y Fl(=)k Fd(l)981 836 y Fm(1)1003 829 y Fd(p)1028 836 y Fi(i)p Fh(\000)p Fm(1)p Fi(;i)737 903 y Fd(l)753 883 y Fh(0)752 916 y Fi(ii)794 903 y Fc(\000)c Fd(p)868 883 y Fh(0)868 916 y Fi(ii)913 903 y Fl(=)k Fd(l)981 910 y Fi(i;i)p Fh(\000)p Fm(1)1074 903 y Fd(p)1099 910 y Fm(1)737 978 y Fd(l)752 985 y Fi(i;i)p Fm(+1)856 978 y Fc(\000)d Fd(p)931 985 y Fi(i;i)p Fm(+1)1037 978 y Fl(=)j Fd(l)1105 985 y Fi(ii)1136 978 y Fd(p)1161 985 y Fm(1)737 1053 y Fd(l)752 1060 y Fi(i)p Fm(+1)p Fi(;i)856 1053 y Fc(\000)d Fd(p)931 1060 y Fi(i)p Fm(+1)p Fi(;i)1037 1053 y Fl(=)j Fd(l)1105 1060 y Fm(1)1127 1053 y Fd(p)1152 1032 y Fh(0)1152 1065 y Fi(ii)1796 829 y Fl(\(6\))60 1211 y(\(The)19 b(last)f(four)g(relations)f(are)h(de\014ned)g(for)g (all)g Fd(i)f Fc(\025)g Fl(2.\))28 b(F)l(or)17 b(eac)o(h)h(p)q(ositiv)o (e)h(in)o(teger)e Fd(n)g Fc(\025)g Fl(2)60 1270 y(w)o(e)e(obtain)g(a)g (\014nite)g(bipartite)f(graph)g Fd(D)q Fl(\()p Fd(n;)8 b(q)r Fl(\))17 b(as)d(follo)o(ws.)21 b(First,)14 b Fd(P)1376 1277 y Fi(n)1418 1270 y Fl(and)h Fd(L)1548 1277 y Fi(n)1590 1270 y Fl(are)g(obtained)60 1330 y(from)23 b Fd(P)31 b Fl(and)24 b Fd(L)p Fl(,)i(resp)q(ectiv)o(ely)l(,)f(b)o(y)f(simply)f (pro)s(jecting)g(eac)o(h)h(v)o(ector)g(on)o(to)g(its)g Fd(n)g Fl(initial)60 1390 y(co)q(ordinates.)e(Let)c(the)f(set)h(of)f(v) o(ertices)g(of)g Fd(D)q Fl(\()p Fd(n;)8 b(q)r Fl(\))18 b(b)q(e)f Fd(P)1169 1397 y Fi(n)1208 1390 y Fc([)12 b Fd(L)1287 1397 y Fi(n)1314 1390 y Fl(.)23 b(Adjacency)17 b(in)g Fd(D)q Fl(\()p Fd(n;)8 b(q)r Fl(\))19 b(is)60 1450 y(no)o(w)f(de\014ned)g(in)h(terms)f(of)i(the)f(\014rst)f Fd(n)t Fc(\000)t 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b(ha)o(v)o(e)d(the)i(same)60 1865 y(girth,)15 b(it)g(follo)o(ws)f(that)i(the)f(graphs)e Fd(C)-5 b(D)q Fl(\()p Fd(n;)8 b(q)r Fl(\))17 b(form)d(a)h(family)g(for)g(whic)o(h)f Fd(\015)i Fc(\025)1583 1845 y Fm(4)p 1583 1854 20 2 v 1583 1882 a(3)1617 1865 y Fl(log)1681 1877 y Fi(q)1703 1865 y Fl(\()p Fd(q)11 b Fc(\000)d Fl(1\).)60 1925 y(With)15 b(few)h(exceptions,)f(these)g(graphs)f(pro)o(vide)g(the)h(b)q(est)h (kno)o(wn)e(asymptotic)h(lo)o(w)o(er)f(b)q(ound)60 1985 y(for)j(the)h(greatest)g(n)o(um)o(b)q(er)d(of)j(edges)f(in)g(graphs)f (of)i(their)f(order)f(and)h(girth.)25 b(Later,)18 b(in)f([13],)60 2053 y(the)k(authors)e(pro)o(v)o(ed)g(that)h(for)h Fd(q)h Fl(o)q(dd,)f(the)g(order)e(of)h Fd(C)-5 b(D)q Fl(\()p Fd(n;)8 b(q)r Fl(\))22 b(is)e(exactly)i(2)p Fd(q)1625 2034 y Fi(n)p Fh(\000b)1705 2019 y Fg(n)p Ff(+2)p 1705 2027 65 2 v 1728 2046 a(4)1775 2034 y Fh(c)p Fm(+1)1846 2053 y Fl(.)60 2112 y(Com)o(bining)d(this)h(with)h(a)g(result)f(from)g ([9])h(on)g(the)g(existence)g(of)g(a)g(girth)g(cycle)g(of)g(length)60 2172 y Fd(n)11 b Fl(+)f(5)16 b(in)g Fd(D)q Fl(\()p Fd(n;)8 b(q)r Fl(\))18 b(for)e(all)g(o)q(dd)g Fd(n)g Fl(and)g(in\014nitely)g (man)o(y)f Fd(q)r Fl(,)h(w)o(e)g(get)h(that)g(the)f(corresp)q(onding)60 2232 y(subfamily)f(of)i(graphs)e Fd(C)-5 b(D)q Fl(\()p Fd(n;)8 b(q)r Fl(\))18 b(satis\014es)d Fd(\015)i Fl(=)994 2212 y Fm(4)p 994 2220 20 2 v 994 2249 a(3)1028 2232 y Fl(log)1093 2244 y Fi(q)1115 2232 y Fl(\()p Fd(q)c Fc(\000)e Fl(1\).)160 2340 y(An)18 b(imp)q(ortan)o(t)e(prop)q(ert)o(y)h (of)h(graphs)e Fd(D)q Fl(\()p Fd(n;)8 b(q)r Fl(\))20 b(and)d Fd(C)-5 b(D)q Fl(\()p Fd(n;)8 b(q)r Fl(\))19 b(is)e(that)i(they)f(con)o(tain)f(a)60 2400 y(sp)q(ecial)d(t)o(yp)q(e)h (of)f(induced)f(subgraph.)19 b(These)14 b(w)o(ere)g(in)o(tro)q(duced)f (b)o(y)h(the)g(authors)f(in)h([11])g(and)948 2520 y(5)p eop 6 5 bop 60 -70 a Fb(the)16 b(electr)o(onic)i(journal)f(of)f(combina)m (torics)h(4)f(\(no.)21 b(2\))16 b(\(1997\))g(,)g(#R13)378 b(6)60 50 y Fl(can)16 b(b)q(e)g(describ)q(ed)f(as)g(follo)o(ws.)21 b(Let)c Fd(R;)8 b(S)16 b Fc(\022)e Fd(GF)7 b Fl(\()p Fd(q)r Fl(\),)17 b(where)e Fc(j)p Fd(R)p Fc(j)f Fl(=)g Fd(r)h Fc(\025)f Fl(1)i(and)f Fc(j)p Fd(S)s Fc(j)e Fl(=)g Fd(s)i Fc(\025)e Fl(1,)60 110 y(and)j(let)697 161 y Fd(P)729 168 y Fi(R)776 161 y Fl(=)e Fc(f)p Fl(\()p Fd(p)p Fl(\))g Fc(2)g Fd(P)1010 168 y Fi(n)1037 161 y Fc(j)p Fd(p)1076 168 y Fm(1)1112 161 y Fc(2)g Fd(R)p Fc(g)697 236 y Fd(L)731 243 y Fi(S)774 236 y Fl(=)g Fc(f)p Fl([)p Fd(l)q Fl(])f Fc(2)h Fd(L)990 243 y Fi(n)1017 236 y Fc(j)p Fd(l)1046 243 y Fm(1)1082 236 y Fc(2)g Fd(S)s Fc(g)p Fd(:)60 334 y Fl(W)l(e)22 b(de\014ne)e Fd(D)q Fl(\()p Fd(n;)8 b(q)r(;)g(R;)g(S)s Fl(\))23 b(to)e(b)q(e)h(the)g(subgraph)d(of)i Fd(D)q Fl(\()p Fd(n;)8 b(q)r Fl(\))23 b(induced)d(on)h Fd(P)1580 341 y Fi(R)1627 334 y Fc([)14 b Fd(L)1708 341 y Fi(S)1737 334 y Fl(.)36 b(F)l(or)60 394 y(\014xed)21 b(\()p Fd(p)p Fl(\))i Fc(2)g Fd(P)359 401 y Fi(R)413 394 y Fl(and)e(arbitrary)f 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Fm(0)1341 1700 y Fc(g)i Fl(and)f Fd(B)h Fl(=)e Fc(f)p Fd(b)g Fl(:)g Fd(b)h Fc(\025)f Fd(b)1799 1707 y Fm(0)1821 1700 y Fc(g)p Fl(,)60 1760 y(for)h(some)g(p)q(ositiv)o(e)g(in)o(tegers)g Fd(a)660 1767 y Fm(0)683 1760 y Fd(;)8 b(b)726 1767 y Fm(0)748 1760 y Fl(.)26 b(W)l(e)17 b(tried)g(to)h(do)g(this)f (constructiv)o(ely)g(b)o(y)g(considering)60 1820 y(sp)q(ecial)f (subgraphs)e(of)j Fd(X)560 1801 y Fi(p;q)631 1820 y Fl(and)f Fd(C)-5 b(D)q Fl(\()p Fd(n;)8 b(q)r Fl(\).)160 1930 y(A)23 b(natural)e(w)o(a)o(y)h(to)h(\014nd)e(a)i Fd(k)r Fl(-regular)d (subgraph)g(in)i(the)h(Ca)o(yley)f(graph)g Fd(X)1683 1937 y Fi(p;q)1760 1930 y Fl(is)g(to)60 1990 y(restrict)16 b(the)h(set)g(of)f Fd(p)c Fl(+)e(1)17 b(generators)e(to)i(a)g (symmetric)e Fd(k)r Fl(-elemen)o(t)g(subset.)22 b(The)16 b(di\016cult)o(y)60 2050 y(is)24 b(to)g(do)g(this)f(in)h(suc)o(h)f(a)g (w)o(a)o(y)h(that)g(the)h(girth)e(of)h(the)g(subgraph)e(equals)i(a)g (prescrib)q(ed)60 2110 y(ev)o(en)f(in)o(teger)e Fd(g)r Fl(.)40 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y(regular)f(graphs)f(of)i(girth)f Fk(at)j(le)m(ast)e Fd(g)i Fl(pro)o(vide)d(upp)q(er)g(b)q(ounds)f(on)i (the)g(orders)f(of)h(\()p Fd(k)r(;)8 b(g)r Fl(\)-cages.)60 938 y(Namely)l(,)60 1049 y Fn(Theorem)22 b(2.1)h([7])d Fk(L)m(et)i Fd(G)g Fk(b)m(e)g(a)g Fd(k)r Fk(-r)m(e)m(gular)g(gr)m(aph)h (of)f(girth)g(at)g(le)m(ast)g Fd(g)h Fk(having)g(the)f(le)m(ast)60 1109 y(numb)m(er)c(of)g(vertic)m(es.)24 b(Then)17 b(the)h(girth)g(of)h Fd(G)f Fk(is)g Fd(g)i Fk(and)e(the)g(diameter)g(of)h Fd(G)f Fk(is)g(at)g(most)g Fd(g)r Fk(.)160 1219 y Fl(A)e(pro)q(of)f(of) h(this)g(theorem)f(can)g(b)q(e)h(found)f(in)h([14,)f(pp.)g(66,)h(384,)f (385],)h(see)f(also)h(the)g(ref-)60 1279 y(erences)f(therein.)21 b(Applying)16 b(it)g(to)g(the)g Fd(k)r Fl(-regular)e(subgraphs)f Fd(C)-5 b(D)q Fl(\()p Fd(n;)8 b(q)r(;)g(R;)g(R)p Fl(\))18 b(of)e Fd(C)-5 b(D)q Fl(\()p Fd(n;)8 b(q)r Fl(\),)60 1339 y(where)17 b Fd(n)f Fl(=)f(2)p Fc(b)358 1317 y Fi(g)q Fm(+1)p 358 1328 V 384 1356 a(2)435 1339 y Fc(c)d(\000)g Fl(5)18 b(and)f 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b(fact)h(that)g(the)g(orders)e(of)h (these)h(constructions)e(actually)i(meet)f(the)h(upp)q(er)60 2017 y(b)q(ound)d(in)h(\(2\))g(for)g Fd(q)i Fl(o)q(dd)e(follo)o(ws)f (from)g([13].)31 b(Therefore)20 b(the)g(constructions)e(w)o(e)i(presen) o(t)60 2077 y(b)q(elo)o(w)f(are)g(in)o(teresting)f(mainly)h(for)g Fd(k)i(<)d(g)r(=)p Fl(2.)30 b(Before)20 b(pro)q(ceeding,)f(w)o(e)g(men) o(tion)g(that)g(in)60 2137 y(ev)o(ery)k(case)f(the)h(graphs)e(w)o(e)h (construct)g(can)g(b)q(e)h(view)o(ed,)h(roughly)l(,)e(as)h(b)q(eing)f (formed)f(b)o(y)60 2197 y(app)q(ending)15 b(an)h(appropriate)f(n)o(um)o (b)q(er)f(of)j(high)f(girth)g(graphs)f(to)i(a)f(\\cen)o(tral")g Fd(g)r Fl(-cycle.)60 2340 y Fn(Case)26 b(1:)39 b Fd(g)28 b Fn(ev)n(en.)42 b Fl(Let)23 b Fd(k)j Fc(\025)e Fl(3)e(b)q(e)h(an)g(in) o(teger,)g Fd(g)j Fl(=)d(2)p Fd(s)i Fc(\025)f Fl(4)e(an)h(ev)o(en)f(in) o(teger,)i(and)60 2400 y Fd(C)17 b Fl(=)d Fd(v)190 2407 y Fm(1)212 2400 y Fd(v)236 2407 y Fm(2)267 2400 y Fd(:)8 b(:)g(:)g(v)357 2407 y Fm(2)p Fi(s)399 2400 y Fd(v)423 2407 y Fm(1)462 2400 y Fl(a)16 b(cycle)h(of)g(order)e Fd(g)r Fl(.)21 b(Let)c Fc(f)p Fd(H)1029 2407 y Fi(ij)1065 2400 y Fc(g)f Fl(b)q(e)h(an)f(arbitrary)f(family)h(of)g Fd(k)r Fl(-regular)948 2520 y(7)p eop 8 7 bop 60 -70 a Fb(the)16 b(electr)o(onic)i(journal)f(of)f(combina)m (torics)h(4)f(\(no.)21 b(2\))16 b(\(1997\))g(,)g(#R13)378 b(8)60 50 y Fl(graphs,)16 b(eac)o(h)g(of)i(order)e Fd(v)j Fl(and)e(girth)f(at)i(least)f Fd(g)r Fl(,)g(1)e Fc(\024)f Fd(i)i Fc(\024)e Fd(s)p Fl(,)k(1)d Fc(\024)f Fd(j)k Fc(\024)d Fd(k)e Fc(\000)e Fl(2.)24 b(In)17 b(eac)o(h)g Fd(H)1825 57 y Fi(ij)60 110 y Fl(c)o(ho)q(ose,)f(arbitrarily)l(,)f(a)h (\\distinguished")e(edge)j(and)f(denote)g(it)h(b)o(y)f Fd(a)1391 117 y Fi(ij)1427 110 y Fd(b)1448 117 y Fi(ij)1483 110 y Fl(.)22 b(No)o(w,)17 b(denote)f(b)o(y)60 169 y Fd(H)105 151 y Fh(\003)101 182 y Fi(ij)155 169 y Fl(the)j(graph)e (obtained)h(from)g Fd(H)754 176 y Fi(ij)808 169 y Fl(b)o(y)h(deleting)f (edge)h Fd(a)1212 176 y Fi(ij)1247 169 y Fd(b)1268 176 y Fi(ij)1303 169 y Fl(.)29 b(Finally)l(,)18 b(form)f(the)i(graph)60 229 y Fd(H)24 b Fl(b)o(y)19 b(adjoining)f(the)i(graphs)e Fd(H)715 211 y Fh(\003)711 242 y Fi(ij)766 229 y Fl(to)i(the)g(cycle)g Fd(C)j Fl(in)c(the)h(follo)o(wing)e(manner:)26 b(F)l(or)19 b(eac)o(h)60 289 y(1)14 b Fc(\024)f Fd(i)h Fc(\024)g Fd(s)p Fl(,)i(1)e Fc(\024)f Fd(j)k Fc(\024)c Fd(k)e Fc(\000)e Fl(2,)16 b(adjoin)e(v)o(ertex)j Fd(a)935 296 y Fi(ij)986 289 y Fl(\(resp)q(ectiv)o(ely)l(,)f Fd(b)1307 296 y Fi(ij)1342 289 y Fl(\))g(of)g(graph)e Fd(H)1618 271 y Fh(\003)1614 302 y Fi(ij)1666 289 y Fl(to)i(v)o(ertex)60 349 y Fd(v)84 356 y Fm(2)p Fi(i)p Fh(\000)p Fm(1)186 349 y Fl(\(resp)q(ectiv)o(ely)l (,)f Fd(v)509 356 y Fm(2)p Fi(i)545 349 y Fl(\))g(of)f Fd(C)t Fl(.)21 b(It)14 b(is)g(trivial)g(to)h(see)f(that)g Fd(H)19 b Fl(is)14 b(a)g Fd(k)r Fl(-regular)e(graph)h(of)h(order)60 408 y Fd(g)f Fl(+)d Fd(s)p Fl(\()p Fd(k)k Fc(\000)d Fl(2\))p Fd(v)18 b Fl(and)e(that)h(the)g(girth)f(of)h Fd(H)k Fl(is)16 b(at)h(most)f Fd(g)r Fl(.)160 517 y(Let)23 b(us)g(sho)o(w)f(that)h Fd(g)r Fl(\()p Fd(H)t Fl(\))j(=)e Fd(g)r Fl(.)41 b(Let)24 b Fd(K)j Fl(b)q(e)c(a)g(cycle)h(in)f Fd(H)k Fl(of)d(order)d(less)i (than)g Fd(g)r Fl(.)60 577 y(Then)15 b Fd(K)k Fl(m)o(ust)14 b(con)o(tain)g(at)h(least)g(one)g(pair)f(of)i(edges)e(of)i(the)f(form)f Fd(a)1380 584 y Fi(i)1394 589 y Ff(0)1414 584 y Fi(j)1430 589 y Ff(0)1453 577 y Fd(v)1477 584 y Fm(2)p Fi(i)1511 589 y Ff(0)1530 584 y Fh(\000)p Fm(1)1584 577 y Fl(,)h Fd(b)1634 584 y Fi(i)1648 589 y Ff(0)1668 584 y Fi(j)1684 589 y Ff(0)1706 577 y Fd(v)1730 584 y Fm(2)p Fi(i)1764 589 y Ff(0)1786 577 y Fl(,)g(as)60 637 y(otherwise)g Fd(K)k Fl(w)o(ould)14 b(b)q(e)i(a)f(cycle)h(in)f(either)g Fd(C)k Fl(or)c Fd(H)1068 644 y Fi(ij)1119 637 y Fl(for)g(some)f Fd(i;)8 b(j)s Fl(.)22 b(But)15 b(no)o(w)g(the)h(p)q(ortion)60 697 y(of)g Fd(K)k Fl(whic)o(h)15 b(lies)g(in)g Fd(H)507 679 y Fh(\003)503 710 y Fi(i)517 715 y Ff(0)537 710 y Fi(j)553 715 y Ff(0)576 697 y Fl(,)g(together)h(with)g(the)g(edge)g Fd(a)1138 704 y Fi(i)1152 709 y Ff(0)1172 704 y Fi(j)1188 709 y Ff(0)1210 697 y Fd(b)1231 704 y Fi(i)1245 709 y Ff(0)1265 704 y Fi(j)1281 709 y Ff(0)1304 697 y Fl(,)f(forms)g(a)h (cycle)g(in)g Fd(H)1732 704 y Fi(i)1746 709 y Ff(0)1766 704 y Fi(j)1782 709 y Ff(0)1820 697 y Fl(of)60 757 y(order)f(less)h (than)g Fd(g)r Fl(.)160 866 y(No)o(w,)d(taking)g(eac)o(h)f Fd(H)583 873 y Fi(ij)632 866 y Fl(to)h(b)q(e)g(the)g Fd(k)r Fl(-regular)e(subgraph)f Fd(C)-5 b(D)q Fl(\()p Fd(g)6 b Fc(\000)t Fl(5)p Fd(;)i(q)r(;)g(R;)g(R)p Fl(\))14 b(of)f Fd(C)-5 b(D)q Fl(\()p Fd(g)6 b Fc(\000)60 925 y Fl(5)p Fd(;)i(q)r Fl(\),)16 b(where)f Fd(q)i Fl(is)e(the)h(smallest)f (o)q(dd)g(prime)f(p)q(o)o(w)o(er)g(for)h(whic)o(h)g Fd(q)h Fc(\025)d Fd(k)j Fl(=)d Fc(j)p Fd(R)p Fc(j)p Fl(,)j(w)o(e)f(obtain)g (the)60 985 y(order)g(of)i Fd(H)k Fl(as)16 b(app)q(ears)g(in)g(\(3\).) 60 1122 y Fn(Case)24 b(2:)34 b Fd(g)26 b Fn(o)r(dd.)33 b Fl(Let)21 b Fd(k)h Fc(\025)e Fl(3)h(b)q(e)g(an)f(in)o(teger,)h Fd(g)h Fl(=)e(2)p Fd(s)14 b Fc(\000)g Fl(1)21 b Fc(\025)f Fl(3)g(an)h(o)q(dd)f(in)o(teger,)h(and)60 1181 y Fd(C)c Fl(=)d Fd(v)190 1188 y Fm(1)212 1181 y Fd(v)236 1188 y Fm(2)267 1181 y Fd(:)8 b(:)g(:)g(v)357 1188 y Fm(2)p Fi(s)p Fh(\000)p Fm(1)450 1181 y Fd(v)474 1188 y Fm(1)512 1181 y Fl(a)17 b(cycle)f(of)h(order)e Fd(g)r Fl(.)21 b(Adjoin)16 b(a)g(new)g(v)o(ertex)h Fd(v)1405 1188 y Fm(2)p Fi(s)1463 1181 y Fl(to)f Fd(v)1547 1188 y Fm(2)p Fi(s)p Fh(\000)p Fm(1)1656 1181 y Fl(but)g(to)g(no)60 1241 y(other)g(v)o(ertex)h(of)g Fd(C)t Fl(.)160 1350 y(With)h Fc(f)p Fd(H)356 1357 y Fi(ij)392 1350 y Fc(g)p Fl(,)h Fd(a)476 1357 y Fi(ij)511 1350 y Fd(b)532 1357 y Fi(ij)568 1350 y Fl(,)g(and)f Fd(H)745 1332 y Fh(\003)741 1363 y Fi(ij)795 1350 y Fl(\(1)f Fc(\024)g Fd(i)h Fc(\024)f Fd(s)p Fl(,)i(1)e Fc(\024)g Fd(j)j Fc(\024)d Fd(k)d Fc(\000)e Fl(2\))19 b(de\014ned)f(exactly)h(as)f(in)60 1410 y(Case)e(1,)h(w)o(e)f (form)g Fd(H)21 b Fl(as)16 b(follo)o(ws:)21 b(F)l(or)16 b(eac)o(h)g(1)e Fc(\024)g Fd(i)g Fc(\024)f Fd(s)f Fc(\000)f Fl(1,)16 b(1)e Fc(\024)g Fd(j)i Fc(\024)e Fd(k)e Fc(\000)f Fl(2,)17 b(adjoin)f(v)o(ertex)60 1470 y Fd(a)86 1477 y Fi(ij)142 1470 y Fl(\(resp)q(ectiv)o(ely)l(,)22 b Fd(b)469 1477 y Fi(ij)504 1470 y Fl(\))f(of)g(graph)f Fd(H)796 1452 y Fh(\003)792 1483 y Fi(ij)848 1470 y Fl(to)h(v)o(ertex)g Fd(v)1092 1477 y Fm(2)p Fi(i)p Fh(\000)p Fm(1)1200 1470 y Fl(\(resp)q(ectiv)o(ely)l(,)h Fd(v)1530 1477 y Fm(2)p Fi(i)1567 1470 y Fl(\))f(of)f Fd(C)t Fl(.)34 b(Next)60 1529 y(for)17 b(eac)o(h)g(1)d Fc(\024)h Fd(j)i Fc(\024)e Fd(k)e Fc(\000)e Fl(3,)18 b(adjoin)e(v)o(ertex)i Fd(a)910 1536 y Fi(sj)967 1529 y Fl(\(resp)q(ectiv)o(ely)l(,)g Fd(b)1290 1536 y Fi(sj)1330 1529 y Fl(\))f(of)h(graph)e Fd(H)1611 1511 y Fh(\003)1607 1542 y Fi(sj)1664 1529 y Fl(to)i(v)o(ertex)60 1589 y Fd(v)84 1596 y Fm(2)p Fi(s)p Fh(\000)p Fm(1)192 1589 y Fl(of)f Fd(C)i Fl(\(resp)q(ectiv)o(ely)l(,)d (v)o(ertex)g Fd(v)778 1596 y Fm(2)p Fi(s)820 1589 y Fl(\).)22 b(Finally)l(,)15 b(adjoin)g Fk(b)m(oth)i Fl(v)o(ertices)f Fd(a)1514 1596 y Fi(s;k)q Fh(\000)p Fm(2)1636 1589 y Fl(and)f Fd(b)1753 1596 y Fi(s;k)q Fh(\000)p Fm(2)60 1649 y Fl(of)i Fd(H)162 1631 y Fh(\003)158 1663 y Fi(s;k)q Fh(\000)p Fm(2)281 1649 y Fl(to)g(v)o(ertex)g Fd(v)517 1656 y Fm(2)p Fi(s)558 1649 y Fl(.)160 1758 y(It)24 b(is)g(routine)f (to)i(c)o(hec)o(k)f(that)h Fd(H)j Fl(is)c Fd(k)r Fl(-regular)e(of)i (order)f Fd(g)18 b Fl(+)e(1)g(+)g Fd(s)p Fl(\()p Fd(k)j Fc(\000)c Fl(2\))p Fd(v)r Fl(,)27 b(and)60 1818 y(that)20 b(the)g(girth)g(of)g Fd(H)k Fl(is)c(at)g(most)f Fd(g)r Fl(.)32 b(One)20 b(uses)f(an)g(argumen)o(t)f(similar)h(to)h(that)g(giv) o(en)f(in)60 1877 y(Case)f(1)g(to)g(establish)f(that)i Fd(g)r Fl(\()p Fd(H)t Fl(\))e(=)f Fd(g)r Fl(.)26 b(T)l(aking)18 b(eac)o(h)g Fd(H)1173 1884 y Fi(ij)1226 1877 y Fl(to)h(b)q(e)f(the)h Fd(k)r Fl(-regular)c(subgraph)60 1937 y Fd(C)-5 b(D)q Fl(\()p Fd(g)6 b Fc(\000)f Fl(4)p Fd(;)j(q)r(;)g(R;)g(R)p Fl(\))15 b(of)f Fd(C)-5 b(D)q Fl(\()p Fd(g)6 b Fc(\000)f Fl(4)p Fd(;)j(q)r Fl(\),)15 b(where)d Fd(q)k Fl(is)d(the)g(smallest)g (o)q(dd)g(prime)f(p)q(o)o(w)o(er)g(for)h(whic)o(h)60 1997 y Fd(q)j Fc(\025)d Fd(k)j Fl(=)d Fc(j)p Fd(R)p Fc(j)p Fl(,)k(w)o(e)f(obtain)g(the)h(order)e(of)i Fd(H)k Fl(as)16 b(app)q(ears)g(in)g(\(4\).)60 2161 y Fn(Remark.)36 b Fl(It)22 b(is)f(easy)h(to)g(see)f(that)h(the)g(diameter)f(of)h(a)f Fd(k)r Fl(-regular)e(graph)i(of)h(order)e Fd(v)k Fl(is)60 2221 y(at)c(least)h(log)309 2233 y Fi(k)q Fh(\000)p Fm(1)393 2221 y Fd(v)h Fl(and)d(that)i(the)f(random)f Fd(k)r Fl(-regular)e (graph)i(has)h(diameter)f(close)h(to)g(this)60 2280 y(lo)o(w)o(er)13 b(b)q(ound,)h(see)h([5,)g(Ch.)f(X3].)21 b(Though)14 b(sev)o(eral)g (explicit)g(constructions)g(of)g(families)g(of)h Fd(k)r Fl(-)60 2340 y(regular)e(graphs)f(with)i(diameters)f(close)h(to)g(log) 961 2352 y Fi(k)q Fh(\000)p Fm(1)1045 2340 y Fd(v)i Fl(are)d(kno)o(wn)h ([5,)g(Ch.)g(X1])g(these)g(all)g(ha)o(v)o(e)60 2400 y(small)k(girth.)29 b(The)20 b(problem)d(of)j(constructing)e(in\014nite)g(families)h(of)g (graphs)f(of)h(large)g(girth)948 2520 y(8)p eop 9 8 bop 60 -70 a Fb(the)16 b(electr)o(onic)i(journal)f(of)f(combina)m (torics)h(4)f(\(no.)21 b(2\))16 b(\(1997\))g(,)g(#R13)378 b(9)60 50 y Fl(and)17 b(small)g(diameter)h(\(i.e.)g(with)g(diameter)f (at)i(most)e Fd(c)8 b Fl(log)1205 62 y Fi(k)q Fh(\000)p Fm(1)1289 50 y Fd(v)r Fl(,)18 b Fd(c)e Fc(\025)h Fl(1)h(a)g(constan)o (t\))g(is)f(far)60 110 y(from)g(trivial.)25 b(F)l(or)17 b(the)h(Raman)o(ujan)d(graphs)h(describ)q(ed)h(in)g(F1,)h(diameter)f (is)g(kno)o(wn)g(to)h(b)q(e)60 169 y(at)d(most)f(2)8 b(log)337 182 y Fi(k)q Fh(\000)p Fm(1)420 169 y Fd(v)h Fl(+)e(2,)15 b(and)f(the)h(pro)q(of)f(of)h(this)f(fact)h(is)f(not)h (simple.)20 b(The)14 b(upp)q(er)g(b)q(ound)f(on)60 229 y(the)i(girth)e(of)i(graphs)e Fd(C)-5 b(D)q Fl(\()p Fd(n;)8 b(q)r Fl(\),)16 b(together)e(with)h(the)f(statemen)o(t)h(in)e(Theorem)h (2.1)g(regarding)60 289 y(diameter)19 b(of)h(cages,)g(implies)e(that)i (for)f(ev)o(ery)h Fd(k)g Fc(\025)f Fl(3)h(there)f(exists)h(a)g(family)f (of)h(graphs)e(of)60 349 y(large)f(girth)g Fc(f)p Fd(G)370 356 y Fi(i)386 349 y Fc(g)411 356 y Fi(i)p Fh(\025)p Fm(1)496 349 y Fl(suc)o(h)f(that)i Fd(g)r Fl(\()p Fd(G)802 356 y Fi(i)819 349 y Fl(\))d Fc(\025)913 329 y Fm(4)p 913 337 20 2 v 913 366 a(3)948 349 y Fl(log)1012 361 y Fi(k)q Fh(\000)p Fm(1)1096 349 y Fd(v)1120 356 y Fi(i)1154 349 y Fl(and)i(diam)o(\()p Fd(G)1418 356 y Fi(i)1435 349 y Fl(\))e Fc(\024)1529 329 y Fm(4)p 1529 337 V 1529 366 a(3)1563 349 y Fl(log)1628 361 y Fi(k)q Fh(\000)p Fm(1)1712 349 y Fd(v)1736 356 y Fi(i)1752 349 y Fl(.)25 b(W)l(e)60 408 y(conjecture)19 b(that)g(the)h(diameter)e(of)h(graphs)f Fd(C)-5 b(D)q Fl(\()p Fd(n;)8 b(q)r Fl(\))20 b(is)f(within)f(an)h (additiv)o(e)f(constan)o(t)h(of)60 468 y(this)d(b)q(ound.)21 b(More)16 b(precisely)l(,)60 608 y Fn(Conjecture.)32 b Fk(Ther)m(e)22 b(exists)f(a)h(p)m(ositive)g(c)m(onstant)g Fd(C)i Fk(such)e(that)f(for)g(al)s(l)g(inte)m(gers)g Fd(n)f Fc(\025)f Fl(2)60 667 y Fk(and)g(al)s(l)e(prime)h(p)m(owers)i Fd(q)r Fk(,)575 787 y Fd(diam)14 b Fl(\()p Fd(C)-5 b(D)q Fl(\()p Fd(n;)8 b(q)r Fl(\)\))16 b Fc(\024)e Fl(\(log)1079 799 y Fi(q)q Fh(\000)p Fm(1)1161 787 y Fd(q)r Fl(\))p Fd(n)d Fl(+)g Fd(C)q(:)948 2520 y Fl(9)p eop 10 9 bop 60 -70 a Fb(the)16 b(electr)o(onic)i(journal)f(of)f(combina)m (torics)h(4)f(\(no.)21 b(2\))16 b(\(1997\))g(,)g(#R13)355 b(10)826 50 y Fn(References)96 233 y Fl(1.)25 b(N.)17 b(Alon,)f(T)l(o)q(ols)g(from)g(Higher)g(Algebra,)g(in:)22 b(Handb)q(o)q(ok)17 b(of)g(Com)o(binatorics,)d(V)l(olume)160 283 y(I)q(I)j(\(edit.)24 b(R.)17 b(L.)g(Graham,)f(M.)g(Gr\177)-25 b(otsc)o(hel,)16 b(L.)i(Lo)o(v\023)-25 b(asz\),)17 b(MIT)g(Press,)f (North-Holland,)160 333 y(New)h(Y)l(ork,)f(1995.)96 437 y(2.)25 b(N.)18 b(L.)h(Biggs,)f(Algebraic)g(Graph)f(Theory)h(\(2nd)g (ed.\),)h(Cam)o(bridge)d(Univ)o(ersit)o(y)h(Press,)160 487 y(1993.)96 592 y(3.)25 b(N.)13 b(L.)f(Biggs)h(and)f(A.)h(G.)g (Boshier,)f(Note)i(on)e(the)h(girth)g(of)g(Raman)o(ujan)d(graphs,)i Fj(Journal)160 641 y(of)k(Com)o(binatorial)e(Theory)j Fl(Series)32 b(B,)16 b(v)o(ol.)22 b(49)16 b(,)h(190-194)f(\(1990\).)96 746 y(4.)25 b(B.)16 b(Bollob\023)-25 b(as,)16 b(Extremal)g(Graph)f (Theory)l(,)h(Academic)g(Press,)g(London,)f(1978.)96 850 y(5.)25 b(B.)16 b(Bollob\023)-25 b(as,)16 b(Random)f(Graphs,)g (Academic)h(Press,)f(London,)h(1985.)96 955 y(6.)25 b(A.)12 b(E.)f(Brou)o(w)o(er,)g(A.)h(M.)g(Cohen,)g(A.)g(Neumaier,)f (Distance-Regular)f(Graphs,)h(Springer-)160 1005 y(V)l(erlag,)16 b(Heidelb)q(erg)g({)g(New)h(Y)l(ork,)g(1989.)96 1109 y(7.)25 b(P)l(.)c(Erd})-25 b(os)21 b(and)h(H.)g(Sac)o(hs,)g(Regul\177) -25 b(are)21 b(Graphen)g(gegeb)q(ener)h(T)l(aillen)o(w)o(eite)f(mit)h (mini-)160 1159 y(maler)12 b(Knotenzahl,)h Fj(Wiss.)f(Z.)h(Univ.)h (Halle)f(Martin)f(Luther)h(Univ.)g(Halle{Witten)o(b)q(erg)160 1209 y(Math.{Natur.Reine)i Fl(12)h(\(1963\),)h(251-257.)96 1313 y(8.)25 b(D.)17 b(A.)i(Holton,)f(J.)f(Sheehan,)g(The)h(P)o (etersen)f(Graph,)g(Australian)g(Mathematical)g(So-)160 1363 y(ciet)o(y)l(,)f(Lecture)h(Notes)g(7,)f(Cam)o(bridge)f(Univ)o (ersit)o(y)h(Press,)f(1993.)96 1479 y(9.)25 b(Z.)18 b(F)q(\177)-26 b(uredi,)17 b(F.)g(Lazebnik,)697 1467 y(\023)691 1479 y(A.)h(Seress,)f(V.)i(A.)f(Ustimenk)o(o,)g(A.)g(J.)g(W)l(oldar,)f (Graphs)g(of)160 1529 y(prescrib)q(ed)h(girth)h(and)g(bi-degree,)f Fj(Journal)g(of)i(Com)o(binatorial)d(Theory)i Fl(Ser.)62 b(B)20 b(64)160 1579 y(\(2\))d(\(1995\),)g(228-239.)71 1683 y(10.)25 b(F.)17 b(Lazebnik,)h(V.)g(A.)g(Ustimenk)o(o,)g(Explicit) f(construction)g(of)h(graphs)f(with)g(arbitrary)160 1733 y(large)d(girth)g(and)g(of)g(large)g(size,)h Fj(Discrete)g(Applied)e (Mathematics)h Fl(60)h(\(1995\),)g(275-284.)71 1838 y(11.)25 b(F.)19 b(Lazebnik,)h(V.)h(A.)f(Ustimenk)o(o,)g(A.)g(J.)f(W)l(oldar,)h (New)g(constructions)f(of)h(bipartite)160 1887 y(graphs)13 b(on)h Fd(m;)8 b(n)14 b Fl(v)o(ertices,)h(with)f(man)o(y)g(edges,)g (and)g(without)h(small)e(cycles,)i Fj(Journal)e(of)160 1937 y(Com)o(binatorial)h(Theory)i Fl(Ser.)21 b(B)c(61)f(\(1\))i (\(1994\),)e(111-117.)71 2042 y(12.)25 b(F.)17 b(Lazebnik,)h(V.)g(A.)g (Ustimenk)o(o,)f(A.)h(J.)f(W)l(oldar,)g(A)h(new)g(series)f(of)h(dense)f (graphs)f(of)160 2091 y(high)f(girth,)h Fj(Bulletin)g(of)h(the)g(AMS)f Fl(32)h(\(1\))g(\(1995\),)g(73-79.)71 2196 y(13.)25 b(F.)14 b(Lazebnik,)h(V.)g(A.)g(Ustimenk)o(o,)g(A.)g(J.)g(W)l(oldar,)f(A)h(c)o (haracterization)f(of)h(the)g(comp)q(o-)160 2246 y(nen)o(ts)g(of)i(the) g(graphs)e Fd(D)q Fl(\()p Fd(k)r(;)8 b(q)r Fl(\),)17 b Fj(Discrete)g(Mathematics)e Fl(157)i(\(1996\))g(271{283.)71 2350 y(14.)25 b(L.)12 b(Lo)o(v\023)-25 b(asz,)13 b(Com)o(binatorial)c (Problems)i(and)h(Exercises,)g(North-Holland,)g(Amsterdam,)160 2400 y(1979.)935 2520 y(10)p eop 11 10 bop 60 -70 a Fb(the)16 b(electr)o(onic)i(journal)f(of)f(combina)m (torics)h(4)f(\(no.)21 b(2\))16 b(\(1997\))g(,)g(#R13)355 b(11)71 50 y Fl(15.)25 b(A.)17 b(Lub)q(otzky)l(,)h(R.)f(Phillips,)e(R.) i(Sarnak,)f(Raman)o(ujan)f(graphs,)g Fj(Com)o(binatorica)g Fl(8)i(\(3\))160 100 y(\(1988\),)f(261-277.)71 199 y(16.)25 b(G.)16 b(A.)h(Margulis,)d(Explicit)j(group-theoretical)e(construction) g(of)i(com)o(binatorial)160 249 y(sc)o(hemes)i(and)h(their)h (application)f(to)h(the)g(design)f(of)h(expanders)e(and)i(concen)o (trators,)160 299 y Fj(Journal)14 b(of)j(Problems)e(of)i(Information)e (T)l(ransmission)e Fl(\(1988\),)k(39-46.)71 399 y(17.)25 b(H.)16 b(L.)g(Mon)o(tgomery)l(,)f(T)l(opics)g(in)h(Multiplicativ)o(e)g (Num)o(b)q(er)f(Theory)l(,)g Fj(Lecture)i(Notes)f(in)160 448 y(Mathematics)f Fl(227,)i(Springer)d(V)l(erlag,)i(New)h(Y)l(ork,)f (1971.)71 548 y(18.)25 b(G.)16 b(Ro)o(yle,)g(Cubic)g(cages,)g(h)o (ttp://www.ss.u)o(w)o(a.edu.)o(au/gor)o(don)o(/cages/in)o(dex.h)o(tml)o (.)71 648 y(19.)25 b(H.)20 b(Sac)o(hs,)g(Regular)e(graphs)h(with)h(giv) o(en)g(girth)f(and)h(restricted)f(circuits,)h Fj(J.)g(London)160 697 y(Math.)h(So)q(ciet)o(y)c Fl(38)g(\(1963\),)f(423-429.)71 797 y(20.)25 b(N.)19 b(Sauer,)g(Extremaleigensc)o(haften)e(regul\177) -25 b(arer)18 b(Graphen)g(gegeb)q(ener)h(T)l(aillen)o(w)o(eite,)f(I)160 858 y(and)c(I)q(I,)h Fj(Sitzungsb)q(eric)o(h)o(te)701 845 y(\177)692 858 y(Osterreic)o(h.)20 b(Acad.)15 b(Wiss.)f(Math.)g (Natur.)h(Kl.)p Fl(,)g(S-B)f(I)q(I,)h(176)160 907 y(\(1967\),)h(9-25;)g (176)h(\(1967\),)f(27-43.)71 1007 y(21.)25 b(H.)17 b(W)l(alther,)f (Eigensc)o(haften)g(v)o(on)h(regul\177)-25 b(aren)15 b(Graphen)h(gegeb)q(ener)g(T)l(aillen)o(w)o(eite)g(und)160 1057 y(minimaler)e(Knotenzahl,)h Fj(Wiss.)h(Z.)h(Ilmenau)e Fl(11)i(\(1965\),)g(167-168.)71 1170 y(22.)25 b(H.)h(W)l(alther,)461 1157 y(\177)455 1170 y(Ub)q(er)g(regul\177)-25 b(are)25 b(Graphen)g(gegeb)q(ener)h(T)l(aillen)o(w)o(eite)g(und)f(minimaler)160 1220 y(Knotenzahl,)15 b Fj(Wiss.)h(Z.)h(T)l(ec)o(hn.)e(Ho)q(c)o(hsc)o (h.)h(Ilmenau)f Fl(11)i(\(1965\),)f(93-96.)71 1319 y(23.)25 b(P)l(.-K.)15 b(W)l(ong,)h(Cages)g({)h(A)g(Surv)o(ey)l(,)e Fj(Journal)g(of)i(Graph)e(Theory)h Fl(V)l(ol.)h(6)f(\(1982\),)h(1-22.) 935 2520 y(11)p eop end userdict /end-hook known{end-hook}if .