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2 Journl of Combintoril Theory, Series B 101 (2011) Contents lists vilble t ScienceDirect Journl of Combintoril Theory, Series B Longest cycles in -connected grphs with given independence number Suil O 1, Dougls B. West 2, Hehui Wu Deprtment of Mthemtics, University of Illinois, Urbn, IL 61801, USA rticle info bstrct Article history: Received 20 April 2010 Avilble online 15 Mrch 2011 Keywords: Fouquet Jolivet Conjecture Circumference Chvátl Erdős Theorem Connectivity Independence number The Chvátl Erdős Theorem sttes tht every grph whose connectivity is t lest its independence number hs spnning cycle. In 1976, Fouquet nd Jolivet conjectured n extension: If G is n n-vertex -connected grph with independence number, nd, theng hs cycle with length t lest (n+ ).Weprove this conjecture Elsevier Inc. All rights reserved. 1. Introduction The fmous theorem of Chvátl nd Erdős [3], published in 1972, reltes connectivity, independence number, nd circumference of grphs. A grph G is -connected if it hs more thn vertices nd every subgrph obtined by deleting fewer thn vertices is connected; the connectivity of G, written κ(g), is the mximum such tht G is -connected. An independent set is set of pirwise nondjcent vertices, nd the independence number of G, written α(g), is the mximum size of such set.thecircumference is the mximum length of cycle in G. Theorem 1.1 (Chvátl Erdős [3]). If G is grph such tht κ(g) α(g), then G hs cycle through ll its vertices. It is nturl to s wht cn be sid when the condition κ(g) α(g) is weened: given -connected n-vertex grph with independence number, where, wht is the best lower bound on the circumference? In 1976, Fouquet nd Jolivet conjectured n nswer. E-mil ddresses: suilo2@mth.uiuc.edu (S. O), west@mth.uiuc.edu (D.B. West), hehuiwu2@mth.uiuc.edu (H. Wu). 1 Reserch prtilly supported by the Koren Reserch Foundtion (MOEHRD, Bsic Reserch Promotion Fund), grnt KRF C Reserch prtilly supported by the Ntionl Security Agency under Awrd No. H /$ see front mtter 2011 Elsevier Inc. All rights reserved. doi: /j.jctb

3 S. O et l. / Journl of Combintoril Theory, Series B 101 (2011) Conjecture 1.2 (Fouquet Jolivet [4]). If G is -connected n-vertex grph with independence number, nd, then G hs cycle with length t lest (n+ ). The cse = simplifies to the Chvátl Erdős Theorem. The conjecture is shrp; infinitely often the circumference of G equls (n + )/. For 2 nd m N, we construct such grph: form G from one copy of the complete grph K nd disjoint copies of K m by ming every vertex in the -clique djcent to ll the other vertices. Now G hs + m vertices, α(g) =, κ(g) =, nd the mximum cycle length is (1 + m). Letting n = + m, we see tht (1 + m) = (n + )/. In 1982, Fournier [5] proved Conjecture 1.2 for { + 1, + 2}. Two yers lter, he lso proved it for = 2 [6], using the fct tht if C 1 nd C 2 re distinct cycles in 2-connected grph G, then there re distinct cycles C 1 nd C 2 in G such tht V (C 1) V (C 2 ) V (C 1 ) V (C 2 ) nd V (C 1 ) V (C 2 ) 2. In 2009, Mnoussis [8] proved the cse = 3 using similr fct. This leds to generl conjecture. Conjecture 1.3 (Chen Chen Liu). If C 1 nd C 2 re distinct cycles in -connected grph G, then there re distinct cycles C 1 nd C 2 in G such tht V (C 1) V (C 2 ) V (C 1 ) V (C 2 ) nd V (C 1 ) V (C 2 ). Recently, Chen, Hu, nd Wu [1] proved Conjecture 1.2 for = 4. In nother pper [2], they proved tht Conjecture 1.3 implies Conjecture 1.2, nd they lso proved Conjecture 1.2 for < 2 1. In this pper, without proving the stronger Conjecture 1.3, we prove Conjecture 1.2 in full. Air Sito sed whether the shrpness construction bove is essentilly unique; possibly our techniques could be used to prove tht. 2. The pth lemm We need nottion for induced subgrphs. Given S V (G), lets = V (G) S. ThesubgrphofG induced by S is the subgrph obtined by deleting the vertices of S; this my be written s G[S] or G S. WhenS ={v}, wewriteg v insted of G {v}. WelsowriteG e for the (non-induced) subgrph obtined by deleting n edge e. The following result of Kouider [7] hs been used in prtil results towrd Conjecture 1.2. Theorem 2.1 (Kouider [7]). If H is subgrph of -connected grph G, then either V (H) cn be covered by cycle in G, or there is cycle C in G such tht α(h V (C)) α(h). A single ppliction of Theorem 2.1 with H = G implies the Chvátl Erdős Theorem (Theorem 1.1) when κ(g) α(g); spnning cycle is gurnteed. When κ(g) <α(g), repetedly pplying Theorem 2.1 with H being the subgrph left by deleting the vertices of erlier cycles shows tht the vertices of grph G cn be covered by t most α(g) κ(g) cycles. For -connected n-vertex grph with independence number, mong these cycles is one of length t lest n/ /, which is close to the conjectured threshold of n/ + (1 /). Inspired by Kouider s result nd her proof, we prove n nlogous theorem bout pths joining two specified vertices. We ctully only need Theorem 2.2 for = 2, but the proof of the generl sttement is the sme. The proof is slight modifiction of Kouider s proof of Theorem 2.1. We will need nottion for subpths of pth. Let u nd v be distinct vertices in grph G. Au, v-pth is pthwithfirstvertexu nd lst vertex v. GivenpthP nd vertices, b V (P), letp[, b] be the, b-pth contined in P. Similrly, let P(, b) = P[, b] {, b}, letp[, b) = P[, b] b, nd let P(, b]=p[, b]. Theorem 2.2. Let G be -connected grph. If H G, nd u nd v re distinct vertices in G, then G contins u, v-pth P such tht V(H) V (P) or α(h V (P)) α(h) ( 1).

4 482 S. O et l. / Journl of Combintoril Theory, Series B 101 (2011) Fig. 1. Finding better pth (Clim 2). Proof. We my ssume tht no u, v-pth P contins V (H). Forechu, v-pth P,letF P be smllest component of G V (P) tht intersects H. Choose u, v-pth P such tht: (i) α(h V (P)) is smllest; (ii) subject to (i), F P hs the fewest vertices. Let p 1,...,p m be the vertices of P (in order) with neighbors in F P.LetU i = V (P(p i, p i+1 )). Clim 1. α(h V (P U i )) > α(h V (P)) for 1 i < m. Otherwise, α(h V (P U i )) = α(h V (P)); ssume this. Let P be u, v-pth obtined from P by deleting U i nd dding p i, p i+1 -pth whose set of internl vertices is nonempty nd lies in F P.IfV (F P ) V (H) V (P ),thenα(h V (P )) < α(h V (P)), contrdicting (i). Hence P does not cover V (F P ) V (H). SinceV (P U i ) V (P ), we hve α(h V (P )) α(h V (P U i )) = α(h V (P)). Since there remins vertex of F P H outside P,wehve V (F P ) < V (F P ), contrdicting (ii). This proves the clim. By Clim 1, restoring U i to the induced subgrph H V (P) increses the independence number. Since U i is nonempty, p 1,...,p m is seprting set, nd hence m. Restoring the vertices of U i in order, strting from p i, let q i be the first vertex t which the independence number increses (see Fig. 1). Tht is, with U i = V (P(p i, q i ]), we hveα(h V (P U )) = α(h V (P)) + 1, but i α(h V (P U i ) q i) = α(h V (P)). Clim 2. For 1 i < j < m, no pth whose internl vertices ll lie outside P joins U i nd U. Otherwise, let j r i U i nd r j U j be the endpoints of such pth ˆP, chosen so tht ri is s close to p i long P s possible. Since F P is component of G V (P), nd vertices of U i nd U j hve no neighbors in F P, the pth ˆP does not visit F P.FormP from P by deleting V (P(p i, r i )) nd V (P(p j, r j )) nd dding ˆP nd p i, p j -pth through F P. Since r j U j, restoring the vertices in P(p j, r j ) to H V (P) does not produce lrger independent set thn exists in H V (P), nd the sme is true of P(p i, r i ). Furthermore, the choice of r i forbids pths from V (P(p i, r i )) to V (P(p j, r j )) in H V (P), so restoring both sets dds them to different components of H V (P), nd hence restoring both does not increse the independence number. We conclude tht α(h V (P )) α(h V (P)). As in the proof of Clim 1, V (F P ) V (H) V (P ) yields strict inequlity nd violtes (i), while V (F P ) V (H) V (P ) nd equlity imply tht P violtes (ii). This proves the clim. Bythechoiceofq i,wehveα(h V (P U i )) α(h V (P)) + 1. Let U = m 1 i=1 U. By Clim 2, i the sets U 1,...,U m 1 lie in different components of G V (P U). Henceα(H V (P U)) α(h V (P)) + m 1. Since α(h) α(h V (P U)) nd m, wehveα(h V (P)) α(h) + 1for the chosen pth P. Theorem 2.2 implies conjecture posed in Chen, Hu, nd Wu [1]. Corollry 2.3. If grph G dmits no vertex prtition (V 1, V 2 ) such tht α(g) = α(g[v 1 ]) + α(g[v 2 ]),then G is connected nd hs no cut-vertex, nd ny distinct vertices u, v V (G) re the endpoints of pth P such tht α(g V (P)) < α(g).

5 S. O et l. / Journl of Combintoril Theory, Series B 101 (2011) Proof. If G is disconnected, then such prtition exists. Suppose tht G is connected nd hs cut-vertex x. Let A be component of G x, nd let B = G x V (A). Let A = G V (B) nd B = G V (A). Ifα(A) = α(a ),then α(g) α ( A ) + α(b) = α(a) + α(b) α(g). Equlity holds throughout, nd (V (A ), V (B)) is the required prtition. The remining lterntive is α(a) = α(a ) 1. Now there is n independent set S of size α(a) tht contins no neighbor of x. We compute α(g) α(a) + α ( B ) = S +α ( B ) α(g), nd (V (A), V (B )) is the required prtition. The finl sttement holds trivilly if G {K 1, K 2 }. Otherwise, Theorem 2.2 now pplies with = 2 nd H = G. The sufficient condition given is not necessry condition, s shown by the union of two complete grphs shring one vertex. Exmples where the conclusion fils include grphs consisting of two disjoint complete grphs plus one edge joining them. 3. Finding good cycle Given disjoint subgrphs F nd H of grph G, let n F, H-pth in G be pth with endpoints in V (F ) nd V (H) nd no internl vertex in V (F ) V (H); this generlizes u, v-pth. Given specified orienttion of cycle C nd vertices, b V (C), letc[, b] be the, b-pth on C in the given orienttion. Similrly, let C(, b) = C[, b] {, b}. Abloc in grph is mximl subgrph hving no cut-vertex; grph is the union of its blocs. Theorem 3.1. Let be n integer greter thn 1. If C is cycle with length t lest in -connected grph G, then for ny nonempty subgrph H of G V (C), there exists cycle C in G such tht V (C) V (C ) V (C) 1 nd α(h V (C )) α(h) 1. Proof. Consider miniml counterexmple H for some grph G nd cycle C. Let L = V (C). If H is disconnected or hs cut-vertex, then α(h) = α(h[v 1 ]) + α(h[v 2 ]) for some prtition (V 1, V 2 ) of V (H), by Corollry 2.3. By the minimlity of H, thereiscyclec in H[V 1 ] such tht V (C) V (C ) (L/) 1 nd α(h[v 1 V (C )]) α(h[v 1 ]) 1. Now α(h V (C )) α(h[v 1 V (C )]) + α(h[v 2 ]) α(h[v 1 ]) 1 + α(h[v 2 ]) = α(h) 1. We my therefore ssume tht H is 2-connected or H {K 1, K 2 }.LetB be the bloc of G V (C) tht contins H. For B, C-pths P 1 nd P 2, define the C-distnce between P 1 nd P 2 to be the distnce in C between the endpoints of P 1 nd P 2 in C. For b V (B), stndrd consequence of Menger s Theorem yields pths from b to C tht pirwise shre only b; cll this b, C-fn. By the pigeonhole principle, the C-distnce between some two pths in b, C-fn is t most L/. Ifb is the only vertex of B (nd hence H = B), then using those two pths to replce the prt of C between their endpoints yields the desired cycle C.Hencewemy ssume V (B) > 1. Let P 1 nd P 2 be two disjoint B, C-pths, with P i hving endpoints u i B nd v i C. Since B is connected nd hs no cut-vertex, Theorem 2.2 gurntees u 1, u 2 -pth P in B such tht α(h V (P)) α(h) 1. If C(v 1, v 2 ) L/ 1, then (C C(v 1, v 2 )) P 1 P P 2 is the desired cycle C (see Fig. 2). Hence we my ssume ( ) the C-distnce between ny two disjoint B, C-pths is more thn L/. Note lso tht B, C-pths with distinct endpoints in B re internlly disjoint, since B is bloc in G V (C). Let c 1,...,c m be the endpoints in C of B, C-pths, indexed so tht c 1,...,c m pper in tht order long fixed orienttion of C. LetP i = C[c i, c i+1 ] (indices modulo m); cll P i the ith segment of C. Let t be the number of indices i (modulo m) such tht c i nd c i+1 re the endpoints of B, C-pths from distinct vertices of B. By( ), ech such segment hs length more thn L/, nd hence t <.

6 484 S. O et l. / Journl of Combintoril Theory, Series B 101 (2011) Fig. 2. A detour to reduce α(h). For b V (B), b, C-fn hs endpoints in C. Some t of the pths long C joining consecutive endpoints of the fn must not contin endpoints of B, C-pths from other vertices of B. Hence these pths re distinct for distinct vertices of B. Consider segment within ech such pth. Since these segments void the t excluded segments, their totl length is less thn L t(l/), which equls L( t)/. For ech vertex of B, choose shortest mong these t segments. The totl length of the union of the chosen segments is less thn L/. Form C from C by deleting the chosen segments nd dding, for ech b B, the two pths in the b, C-fn whose endpoints re the ends of the segment chosen for b (see Fig. 3). The subgrph C is cycle, becuseb, C-pths from distinct vertices of B re internlly disjoint. Since the totl length of the chosen segments is less thn L/ nd V (H) V (B) V (C ), the cycle C hs the desired properties. Fig. 3. Sipping the chosen segments. Lemm 3.2. If G is -connected grph with independence number, nd 0 l, then there exist cycles C 0,...,C l stisfying the following conditions: (1) α(g l i=0 V (C i)) l, (2) V (C i ) i 1 j=0 V (C j) V (C 0) 1 for 1 i l. Proof. We prove the clim by induction on l. Forl = 0, Theorem 2.1 with H = G provides cycle C 0 such tht α(g V (C 0 )). For the induction step, consider l with 0 < l, nd suppose tht cycles C 0,...,C l 1 exist stisfying the clim for l 1. We observe first tht V (C 0 ) ; when l = 1 this holds becuse the cse l = 0 of (1) sttes tht α(g V (C 0 )), nd when l > 1itholds becuse the left side of (2) is nonnegtive. Let H = G l 1 i=0 V (C i); by hypothesis, α(h) (l 1). We my ssume α(h) 1; otherwise, just let C l = C 0.Since V(C 0 ), we cn pply Theorem 3.1 using C 0 s C to obtin cycle C in G such tht V (C 0 ) V (C ) V (C 0) 1 nd α(h V (C )) α(h) 1 l. Now dding C to the list s C l stisfies (1), but we must lso stisfy (2). Cse 1: V (C ) V (C 0 ). Note tht ( V C ) l 1 j=0 V (C i ) ( V C ) V (C 0 ) V (C0 ) ( V C ) V (C 0) 1. In this cse it suffices to dd C s C l. Cse 2: V (C ) > V (C 0 ). Define new list C 0,...,C of cycles by letting C l 0 = C nd letting C i = C i 1 for 1 i l. Nowα(G l i=0 V (C i )) = α(h V (C )) l, stisfying (1). Also, for

7 S. O et l. / Journl of Combintoril Theory, Series B 101 (2011) i = 1 we hve V (C i ) i 1 j=0 V (C j ) = V (C 1 ) V (C 0 ) = V (C 0) V (C ), nd for 2 i l we hve V (C i ) i 1 j=0 V (C j ) V (C i 1) i 2 j=0 V (C j). Inbothcses, ( V C i ) i 1 j=0 ( V C ) j V (C 0) 1 V (C 0 ) 1. Hence C 0,...,C l stisfies the required conditions. We cn now prove Conjecture 1.2, the conjecture of Fouquet nd Jolivet. Corollry 3.3. If G is -connected n-vertex grph with independence number, nd, then G hs cycle of length t lest (n+ ). Proof. Consider l = in Lemm 3.2. By (1), the resulting cycles C 0,...,C l cover V (G). Usingthis nd then summing the inequlities in (2), we obtin n = V (C0 ) l i 1 + V (C i) V (C j ) V (C0 ) ( ) V (C0 ) + ( ) 1. i=1 j=0 The inequlity simplifies to V (C 0 ) (n+ ). References [1] G. Chen, Z. Hu, Y. Wu, Circumferences of -connected grphs involving independence numbers, mnuscript, [2] G. Chen, Z. Hu, Y. Wu, Circumferences of -connected grphs involving independence numbers, II, mnuscript, [3] V. Chvátl, P. Erdős, A note on Hmiltonin circuits, Discrete Mth. 2 (1972) [4] J.L. Fouquet, J.L. Jolivet, Probléme 438, in: Problémes combintoires et théorie des grphes, Univ. Orsy, Orsy, [5] I. Fournier, thesis, University Prix-XI, Orsy, [6] I. Fournier, Longest cycles in 2-connected grphs of stbility α, in: B. Alspch, C.D. Godsil (Eds.), Cycles in Grphs, Elsevier, North-Hollnd, [7] M. Kouider, Cycles in grphs with prescribed stbility number nd connectivity, J. Combin. Theory Ser. B 60 (1994) [8] Y. Mnoussis, Longest cycles in 3-connected grphs with given independence number, Grphs Combin. (2009)