ON THE CHROMATIC NUMBER OF GENERALIZED STABLE KNESER GRAPHS

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1 ON THE CHROMATIC NUMBER OF GENERALIZED STABLE KNESER GRAPHS JAKOB JONSSON Abstract. For each teger trple (, k, s) such that k 2, s 2, a ks, efe a graph the followg maer. The vertex set cossts of all k-subsets S of Z such that ay two elemets S are o crcular stace at least s. Two vertces form a ege f a oly f they are sjot. For the specal case s = 2, we get Schrjver s stable Keser graph. The geeral costructo s ue to Meuer, who cojecture that the chromatc umber of the graph s s(k 1). By a famous result ue to Schrjver, the cojecture s kow to be true for s = 2. The ma result of the preset paper s that the cojecture s true for s 4, prove s suffcetly large terms of s a k. The proof techques o ot apply to the case s = 3, whch remas early completely ope. 1. Itroucto a summary Let be a teger, a let Z eote the set of cogruece classes of tegers moulo. For a teger x, we let x eote the correspog cogruece class moulo ; the moulus wll always be clear from cotext. For a oempty subset S Z a a teger x such that x S, efe σ(x; S) to be the smallest y > x such that y S. Let s be a postve real umber. A set A S s s-sparse S Z f, for each a such that a A a each 1, we have that (1) σ (a; A) σ s (a; S). If S = Z, the (1) smplfes to (2) σ (a; A) a s. Assumg that s s a teger, a vewg the elemets of S as arrage creasg orer arou a crcle, the set A s s-sparse exactly whe there are at least s 1 elemets S betwee ay two elemets A. For ay S Z a ay real umber s 1, let SG s S,k be the graph whch the vertces are all s-sparse k-subsets of S, a two vertces form a ege f a oly f they are sjot. We wrte SG s,k = SGs Z,k. Choosg s = 1, we obta the Keser graph KG,k [3, 4], whereas s = 2 yels Schrjver s stable Keser graph [6]. For a loopless graph G a a teger r 2, let G (r) be the r-uform hypergraph o the same vertex set as G whch the eges are all clques of sze r G. For a teger c 1, a proper c-colorg of a hypergraph H s a fucto h : V (H) {1,..., c} such that o ege s moochromatc; h(τ) cossts of at least two elemets 1991 Mathematcs Subject Classfcato. 05C15. Key wors a phrases. Chromatc umber, stable Keser graph. Supporte by the Swesh Research Coucl (grat ). 1

2 2 JAKOB JONSSON for each ege τ. Let χ(h) be the smallest c such that there exsts a proper c-colorg of H. For a graph G, we wrte χ (r) (G) = χ(g (r) ). Meuer suggeste the followg cojecture, whch geeralzes several well-kow theorems a cojectures. Cojecture 1.1 (Meuer [5]). Let k,, r, s be tegers such that k 2, s r 2, a ks. The s(k 1) χ (r) (SG s,k ) =. The cojecture extes a smlar cojecture ue to Alo, Drewowsk, a Luczak [1] for the case r = s. Oe may exte the cojecture further by lettg s be ay real umber satsfyg s 2. Oe recto of the cojecture s easy. Specfcally, we have the followg upper bou. Proposto 1.2. Let s be a real umber, a let k,, r be tegers such that k 2, s r 2, a ks. The s(k 1) χ (r) (SG s,k ). Proof. Throughout ths proof, etfy Z wth the set {1,...,}. To ay vertex S SG s,k, assg the color (m S)/(). Note that ay vertex wth a gve color x must have a oempty tersecto wth the set {k : (x 1)() < k x()}. Sce ths set has sze r 1, there s o moochromatc r-clque SG s,k. Choosg = k 1 (2), we get that o vertex SG s,k s cotae the set { : s(k 1) + 1 }. As a cosequece, we are oe. Meuer state a prove Proposto 1.2 the case that s s a teger; the proof s etcal to the oe above. Earler, Alo, Drewowsk, a Luczak obtae the same bou the specal case r = s. By Proposto 1.2, to obta Cojecture 1.1 for gve values k,, r, s, t suffces to prove that s(k 1) χ (r) (SG s,k ). I the preset paper, the focus s o orary graphs, a the ma result reas as follows. Theorem 1.3. Let s be a real umber, a let t be a postve teger such that 2 t s/2. Defe = t/s. Wheever ks, we have that χ(sg s,k) I partcular, f χ(sg t,k) t(k 1), the χ(sg s,k) s(k 1) χ(sgt,k). t/s t/s s(k 1). t/s

3 ON THE CHROMATIC NUMBER OF GENERALIZED STABLE KNESER GRAPHS 3 See Secto 2 for the proof of Theorem 1.3. It remas a ope problem whether t s possble to exte Theorem 1.3 to the case s/2 < t s. Theorem 1.3 oes ot rema true geeral f t(k 1) s a o-teger; see Secto 1.2. It s plausble that the theorem oes rema true the case that t(k 1) s a teger, but our proof oly works whe t tself s a teger. The ea of the proof s to pck a (/)-sparse subset S Z of sze a the show that t-sparse subsets of S are s-sparse Z. Ths s typcally false for o-tegers t. Explag the ea of the proof a bt more, we wll coser the sets S g = S g, where g Z. Gve a optmal proper colorg of SG s,k, we wll euce that each SG t S g,k s a subgraph of SG s,k requrg at least χ(sg t,k) colors. Examg how the sets S g tersect, we wll the observe that each color s use o at most of the subgraphs SG s,k. The cocluso wll be that χ(sg t,k) χ(sg s,k) Cosequeces of the ma result. To escrbe some of the cosequeces of Theorem 1.3, let us revew what s kow about Cojecture 1.1. Usg topologcal methos, Lovász [4] showe that χ(kg,k ) = 2(k 1) wheever k 2 a 2k. Shortly after, Schrjver [6] stregthee ths result, showg that the chromatc umber remas the same for the subgraph SG 2,k. Theorem 1.4 (Schrjver [6]). For k 2 a 2k, we have that χ(sg 2,k) = 2(k 1). Thaks to Theorem 1.4, we may euce the followg from Theorem 1.3. Corollary 1.5. Let q 1 a k 2 be tegers. Wheever 2 q k, we have that χ(sg 2q,k) = 2 q (k 1). Proof. Let q 2, a assume by ucto that χ(sg 2q 1,k ) = 2 q 1 (k 1) for 2 q 1 k. Choosg s = 2 q a t = 2 q 1 Theorem 1.3, we get that for 2 q k. Now, χ(sg 2q,k) 2 q (k 1) /2 /2 2 q (k 1) = /2 /2 /2 2 q (k 1) /2 { 0 f s eve, f s o, 2 q (k 1) 1 whch s strctly less tha 1 both cases; 2 q k. By Proposto 1.2, we are oe. Corollary 1.6. Let s 4 be a real umber, a let q = log 2 (s/2). Wheever ks a k 2, we have that χ(sg s,k) s(k 1) 2q /s 2 q /s 2 q s(k 1). /s I partcular, for each real umber s 4 a each teger k 2, we have that χ(sg s,k) = s(k 1) wheever { s } max 2 q ǫ (s(k 1) + ǫ), ks, where ǫ = s(k 1) + 1 s(k 1).

4 4 JAKOB JONSSON Proof. We obta the frst statemet of the corollary by choosg t = 2 q Theorem 1.3 a applyg Corollary 1.5 wth = 2 q /s stea of. Corollary 1.5 oes apply, because 2 q (ks)/s = 2 q k = 2 q k. For the last statemet, ote that χ(sg s,k) = s(k 1) as soo as (3) 2 q /s 2 q /s 2 q s(k 1) < ǫ. /s Sce 2 q /s > 2 q /s 1, we obta that (3) s true as soo as Ths coclues the proof. 1 s 2 q s(k 1) ǫ /s 1 2 q (s(k 1) + ǫ). ǫ As a se remark, ote that ǫ = 1 wheever s(k 1) s a teger. The restrcto s 4 Corollary 1.6 s because of the restrcto t s/2 Theorem 1.3. For 2 < s < 4, very lttle seems to be kow. Usg computer, Meuer [5] has establshe Cojecture 1.1 for (r, s) = (2, 3) the case that 2k + 5 a also the case that (, k) = (14, 4) Some remarks. Theorem 1.3 oes ot rema true geeral f t s a oteger. For example, assume that t > 4 s a ratoal umber such that t(k 1) s ot a teger, a assume that s(k 1) s a teger. Let have the property that = t/s s a teger. Choosg, a hece, large eough, Cojecture 1.1 s true for SG t,k; apply Corollary 1.6. We get that χ(sg s,k ) χ(sgt,k ) s(k 1) t(k 1) s(k 1) t(k 1) < (k 1)(t s) = = 0; the seco equalty s strct, as we assume that t(k 1) s ot a teger. Cojecture 1.1 (extee to real s) mples Theorem 1.3 for ay s t 2 such that t(k 1) s a teger. Namely, assumg the cojecture s true for SG s,k, we get that χ(sg s,k ) χ(sgt,k ) s(k 1) t(k 1) = (k 1)(t s) Hypergraphs. Let us also revew the stuato for hypergraphs. Throughout ths secto, all parameters are tegers. Aga usg topologcal methos, Alo, Frakl, a Lovász [2] extee the result of Lovász [4] to Keser hypergraphs, provg that r(k 1) χ (r) (KG,k ) = wheever k 2, r 2, a kr. Regarg stable Keser hypergraphs, Alo, Drewowsk, a Luczak [1] maage to settle Cojecture 1.1 the partcular case that r = s = 2 q for some teger q.

5 ON THE CHROMATIC NUMBER OF GENERALIZED STABLE KNESER GRAPHS 5 Theorem 1.7 (Alo, Drewowsk, a Luczak [1]). Let q 1 a k 2. For every 2 q k, we have that 2 χ (2q) q (SG 2q,k ) = (k 1) 2 q. 1 Meuer [5] prove the followg result, exteg a smlar result ue to Alo, Drewowsk, a Luczak [1]. Theorem 1.8 (Meuer [5]). Suppose that the followg hol for gve parameters r 1 2 a s 2 r 2 2. Cojecture 1.1 s true for (r, s) = (r 1, r 1 ) a all a k such that kr 1 a k 2. Cojecture 1.1 s true for (r, s) = (r 2, s 2 ) a all a k such that ks 2 a k 2. The Cojecture 1.1 s true for (r, s) = (r 1 r 2, r 1 s 2 ) a all a k such that ks a k 2. The followg result s a cosequece of Corollary 1.5, Theorem 1.7, a Theorem 1.8. Corollary 1.9. Let p a q be ay postve tegers such that p q, a let k 2. The 2 χ (2p) q (SG 2q,k ) = (k 1) 2 p. 1 for 2 q k. Proof. The case p = 1 s Corollary 1.5. For p 2, use Theorem 1.8 wth r 1 = 2 p 1 a (r 2, s 2 ) = (2, 2 q p+1 ); apply Theorem 1.7 a Corollary Proof of Theorem 1.3 Let s 4 be a real umber, a let t be a teger such that 2 t s/2. For a gve teger, wrte = t/s. Let { } S = / : 0 1 Z. Lemma 2.1. If a set A S s t-sparse S, the A s s-sparse Z. I partcular, SG s,k cotas SGt S,k as a subgraph. Proof. Assume that A s t-sparse S. Coser a elemet a 0 Z such that a 0 A. For m 1, wrte a m = σ m (a 0 ; A). We wat to prove that a m a 0 ms. We have that a m = σ l (a 0 ; S) for some l > 0. By assumpto, l mt. Now, a 0 = / for some, a a m = j/ for some j, whch meas that l = j. We get that j j a m a 0 = l mt = = ms, t/s whch coclues the proof.

6 6 JAKOB JONSSON Let S g = S g for g Z. Defe a graph G(, S) wth vertex set Z a wth a ege betwee g a h wheever S g S h =. Whle ot useful what follows, oe may observe that G(, S) = SG /, ; ths s because S a ts traslates are the oly (/)-sparse -subsets of Z. A vertex set I a graph G s epeet f o two vertces I are ajacet G. The epeece umber of G s the greatest value a such that there s a epeet set G of sze a. Lemma 2.2. The epeece umber of G(, S) s. Proof. Let I be a epeet set G(, S). By cyclc symmetry, we may assume that 0 I. Suppose that p I. The S 0 S p, whch s true f a oly f there are a j such that Ths mples that / = j/ p p = j/ /. p = (j )/ or p = (j )/ + 1. We coclue that I {g, g + 1 : g S}. Now, / s/t 2, whch mples that S g S g+1 =. I partcular, at most oe of g a g + 1 belogs to I for each g S; hece I S =. To see that the epeece umber s, ote that the set S forms a epeet set G(, S); S g cotas the elemet 0 for each g S. Proof of Theorem 1.3. Coser a proper colorg of SG s,k wth χ(sgs,k ) colors. For each color, let C eote the set of elemets g Z such that some vertex of SG t S g,k s gve the color. By Lemma 2.1, SG t S g,k s cotae SG s,k, whch mples that (4) C g Z χ(sg t S g,k) = χ(sg t S,k). Now, each C s a epeet set the graph G(, S), because the colorg s proper. By Lemma 2.2, we get that C, whch yels that (5) C χ(sg s,k ). Combg (4) a (5), we obta the theorem Remark. I the proof of Theorem 1.3, we cosere the subgraphs SG t S g,k of the graph SG s,k. We efe a graph G(, S) a observe that χ(sg t S ) α(g(, S)) g,k χ(sgs,k ), g Z where α(g(, S)) eotes the epeece umber of G(, S). Ths s a specal case of a more geeral fact, whch we state for completeess. Let H = (V, E) be a graph, a let V 1,..., V be subsets of V. Defe a graph G wth vertex set {1,...,} a wth a ege betwee a j wheever the complete bpartte graph wth blocks V a V j s a subgraph of H. The χ(h[v ]) α(g) χ(h), =1

7 ON THE CHROMATIC NUMBER OF GENERALIZED STABLE KNESER GRAPHS 7 where H[V ] s the uce subgraph of H o the vertex set V. To see ths, coser a optmal colorg of H. For each color, let C be the set of ces j such that some vertex V j s gve the color. As the proof of Theorem 1.3, we euce that C χ(h[v ]). Moreover, each C s a epeet set G. Namely, f a a b are ajacet G, the all x V a are ajacet to all y V b. We coclue that C α(g) χ(h). Refereces [1] N. Alo, L. Drewowsk, a T Luczak, Stable Keser hypergraphs a eals N wth the Nkoým property, Proc. Amer. Math. Soc. 137 (2009), [2] N. Alo, P. Frakl, a L. Lovasz, The chromatc umber of Keser hypergraphs, Tras. Amer. Math. Soc. 298 (1986), [3] M. Keser, Aufgabe 360, Jahresber. Deutsch. Math.-Vere. 58 (1955), 2. Abtelug, 27. [4] L. Lovász, Keser s cojecture, chromatc umber, a homotopy, J. Combatoral Theory 25 (1978), [5] F. Meuer, The chromatc umber of almost stable Keser hypergraphs, J. Comb. Theory Ser. A, Accepte. [6] A. Schrjver, Vertex-crtcal subgraphs of Keser graphs, Neuw Arch. Wsk. (3) 26 (1978), o. 3, Departmet of Mathematcs, KTH Royal Isttute of Techology, Stockholm, Swee E-mal aress: jakobj@kth.se