Ramsey theory and chromatic numbers

Pacific Journal of Mathematics RAMSEY THEORY AND CHROMATIC NUMBERS G ARY T HEODORE C HARTRAND AND A LBERT DAVID P OLIMENI Vol. 55, No. 1 September 1974 P AC I F I C JOU R N AL OF M AT H E M AT I C S Vol. 55, N o. 1, 1974 RAMSEY TH EORY AN D CH ROM ATIC N U MBERS G ARY C H ARTRAN D AN D ALBE R T D . P OLI M E N I LetzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA χ ( G) denote the chromatic number of a graph G. For positive integers nίf n2, , nk (k ^ 1) the chromatic Ramsey number χ ( n ίf n2, , nk) is defined as the least positive integer p such that for any factorization K p — U*= i Gif χ { G t) Ξ> w* for at least one i,l ^i ^k. It is shown that x(nu n2, , nk) = 1 + ΓR= i (nt — 1). The vertex- arboricity a(G) of a graph G is the fewest number of subsets into which the vertex set of G can be partitioned so that each subset induces an acyclic graph. For positive integers nlt n2, , w* (ft ^ 1) the vertexarboricity Ramsey number a(nltn2, - ,nk) is defined as the least positive integer p such that for any factorization K p — U*=i Gi> d(Gt) ^ Ύ ii for at least one i, 1 rg % <Ξ k. It is shown , nk) = 1 + 2k ΓR= i (nt —1). that a(nlt n2, I n t r o d u c t io n * The classicalzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK Ramsey number r{m, ri), for positive integers m and n, is t h e least positive in teger p such t h a t for an y graph G of order p, either G contains t h e complete graph Km of order m as a subgraph or t h e complement G of G contains Kn as a subgraph . More gen erally, for k(^ 1) positive in tegers nfί n2, , %, the Ramsey number r(nlf n2, • • • , nk ) is defined as t h e least positive U Gk (i.e., integer p such t h a t for an y factorization Kv — G1\ J G2\ J the Gi are spanning, pair wise edge- disjoint, possibly em pty subgraphs of Kp such t h a t t h e union of t h e edge sets of t h e G, equals the edge set of KP), Gi contains Kn. as a subgraph for a t least one ,ί 1 ^ i ^ k. I t is known (see [5]) t h a t all such R am sey n um bers exist; however, th e actual values of r(nl9 n2, , nk ), k ^ 1, are known in only seven cases (see [2, 3]) for which min {nlf n2, , nk } ^ 3. A clique in a graph G is a m axim al complete subgraph of G. The cϊΐgwe number (ω G) is t h e m axim um order am ong t h e cliques of G. The Ramsey num ber r{nu n2, , nk ) m ay be altern atively defined as t h e least positive in teger p such t h a t for an y factorization Kv = UGfe, ω ίG J ^ w, for a t least one i, l^ί<,k. ftU ^U The foregoing observation su ggest s t h e following definition. Let / be a graphical param eter, and let nu n2, , nkf k^l be positive in tegers. The f- Ramsey number f(nlf n2, , nk ) is t h e least positive integer p such t h a t for an y factorization Kp = G1 (J G2 (J U Gk , f(Gi) ^ Π i for a t least one i, 1 <Z i <Z k. H ence, (ω n u n2, , nk ) = KΛi> %2, • • • , ^fc)> i e., t h e ω- Ramsey n um ber is t h e Ramsey number. The object of th is paper is to in vestigate / - Ramsey numbers for two graphical param et ers / , namely ch rom atic n um ber and vertexarboricity. 39 40 G. CHARTRAN D AN D A. D. POLIMEN I C h ro m at ic R am sey n um bers* ThezyxwvutsrqponmlkjihgfedcbaZYXWVUTSR chromatic number χ( G) of a graph G is t h e fewest num ber of colors which may be assigned to t h e vertices of G so t h a t adjacent vertices are assigned different colors. F or positive in tegers nlt n2, - —,nk , t h e chromatic Ramsey number χ( n yί n2, , nk ) is t h e least positive integer p such t h a t for any factorization Kp = Gx U G2 U Gk , χ{G %) ^ nt for some i, 1 <^ i g; k. The existence of t h e n um bers χ( n u n2, • • • , nk ) is guaran teed by t h e fact t h a t χ( n fί n2, , nk ) ^ r( ^ x , n2, , wA). We are now preWe begin with a pared to presen t a formula for χ( n lt n2, - - - ,nk). lemma. If G = G x u G 2 U LE M M A. U Gk , then P roo/ . F or i = 1, 2, , k, let a χ(G t )- coloring be given for Gt. We assign to a vertex v oΐ G t h e color (c lf c2, • • • , cfe), wh ere ct is t h e color assigned to v in Gt. This produces a coloring of G using a t most IK«iZ(G<) colors; hence, χ(G ) ^ Π t TH EOREM 1. For positive integers nlf n2, lt , nk , n2, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH . , nk) = 1 + Π (nt - 1) . l Proof. The result is immediate if ^ = 1 for some ϊ; hence, we assume that ^ ^ 2 for all i, 1 ^ ΐ ^ Λ. F irst, we verify that X(nlt n2, - *,nk)^ Π Let p = 1 + Π i= i (^ί ~ 1)> and assum e t h ere exists a factorization if* = G 1U G 2 U U Gk such t h a t %((?*) ^ ^ - 1 for each i = 1,2, • • - ,&. Then by t h e Lemma, it follows t h a t 1 + Π (*« - 1) = X(K,) ^ Π Z(G.) ^ Π (nt <=1 ί= l 1) , i= l which produces a contradiction. Th us, in an y factorization Kp = (?i U G 2 U U Gk for j> = 1 + Π?= i (nt - 1), we have χ{G τ ) ^ ^ f for at least one i, 1 <* i <^ k. In order to show t h a t k X(nlf n2, we exhibit a factorization , nk ) ^ 1 + Π (nt ^ f c = Gx U (?2 U 1) , U Gk9 where Nh = RAMSEY THEORY AND CHROMATIC N UMBERS 41 Πl= i (% — 1) and zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA χ( Gi) <; nt — 1 for i = 1, 2, , k. The factorization is accomplished by employing induction on k. For k = 1, we simply observe that χ( K N ) — χ( K ni_^ —nt — 1. Assume there exists a factorization KNk _ l = JE?!U JBΓ U U - fiffe.i such th at χ(H"<) ^ ^ —1 for i = 1, 2, • • • ,& — !. Let JP denote % — 1 (pairwise disjoint) copies of KNk χ and define Gk by G& = F . Thus, Gk contains nk - 1 pairwise disjoint copies of H t for i = 1, 2, , k — 1, which we denote by G> Hence, i^ f c = G x U G 2 U U Gk , where χ( G %) ^ nt — 1 for each ,ί 1 g ΐ ^ ί;, which produces the desired result. 2 Vertex- arboricity Ramsey numbers* The vertex- arboricity a{G) of a graph G is the minimum number of subsets into which the vertex set of G may be partitioned so th at each subset induces an acyclic subgraph. As with the chromatic number, the vertex- arboricity may be considered a coloring number since a(G) is the least number of colors which may be assigned to the vertices of G so that no cycle of G has all of its vertices assigned the same color. Our next result will establish a formula for the vertex- arboricity Ramsey number a(nu n2, , nk ) f defined as the least positive integer p such that for every factorization KP = (?! U G2 U U Gk9 a(Gt) ^ nt for , some i, 1 <£ i <£ ifc. Since a(Kn) = {n/ 2}, it follows that a(nlf n2, %) ^ r(2nt — 1, 2n2 — 1, , 2nk — 1). In the proof of the following result, we shall make use of the (edge) arboricity α x(G) of a graph, which is the minimum number of subsets into which the edge set of G may be partitioned so that the subgraph induced by each subset is acyclic. I t is known (see [1, 4]) that aJJK,) — {n/ 2}. THEOREM 2 . For positive integers n lf n2, ' " , n k , a(nlf n2, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR . , nk) = 1 + 2k Π fa< - 1) . Proof. In order to show th at k <Φ i, n2, , nk ) ^ 1 + 2k Π (nt - 1) , we let p = 1 + 2k Πi= i (n{ — 1) and assume th ere exists a factorization Kp = G1 U G2 U U Gk such t h a t a(G^) <. nt — 1 for each i = 1, 2, • • • ,&. F or each i —1, 2, , Λ, th ere isa partition {£/",,!, Ϊ7<fϊ, , t7ifW<_J of th e vert ex set V(Gt) of G f such t h a t t h e subgraph {Uii5} of Gi induced by Uifj is acyclic, j = 1,2, , ^ έ — 1. At least one of the sets U1Λ , Ult2, , U1%%1- 19 say UUmι , contains a t least 1 + 2k Πi=2 (w>i — 1) vertices. Thus, a t least one oft h e sets UttU U2}2, , 42 G. CHARTRAND AN D A. D . POLIMEN I ίf2,tt2- i> sayzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB U2>m2y contains a t least 1 + 2ft Π Ls (^i — 1) vertices of U1>mi. P roceeding inductively, we arrive a t subsets i7lfWl, U2>m2, • • • , Ukίm]e such t h a t f|l= i Ui)m. contains a t least 1 + 2ft Π t ί + i i^i —1) vertices, 1 ^ ί <£ ft —1. I n particular, Π L i Ui>m., contains a set Uhaving 1 + 2k vertices. F or each ί = 1, 2, « ,ft, <ί7> is an acyclic This implies t h a t αx(ίΓ1+sjfe) ^ ft, which subgraph of t h e graph (Ui>mi). is con tradictory. Therefore, &((?*) ^ nt for a t least one i, 1 ^ i ^ k. The proof will be complete once we have verified t h a t k a(nl9 n2, , nk ) ^ 1 + 2k Π ( ^ - 1) • = 11 Let r = Π ί= i (^i ~" 1) We shall exhibit a factorization K2kr = G1\ J U Gk such t h a t a(Gt) ^ w, - 1 for i = 1, 2, , &. We begin G2 U with r pair wise disjoint copies of K2k , labeled K2\ , Kikf , K2k . Since ^i(^2&) = ^> it follows t h a t K2k — (Ji= i Fif where each Ft is an acyclic grap h . We in troduce t h e n otation Fit to denote th e Ft contained in Kι 2k , 1 = 1,2, - - , r an d i = 1, 2, • • • ,&. With each of t h e r Λ- tuples (clf c2, , ck ), cά = 1, 2, , % —1 an d i = 1, 2, , ft, we identify a complete graph K\ k , I = 1, 2, , r, in such a way t h a t t h e identification is one- to- one. Then, for each i = 1, 2, , ft and I = 1,2, , r, we associate with ί7^ t h e ft- tuple identified with K2\ . Define t h e graph Gif i = 1, 2, , ft, t o consist of t h e graph s Filf Fi2, ,i^r; in addition, each vertex of Fis is adjacen t to each vertex of Ftί , s, t = 1, 2, , r, provided t h e ΐ t h coordinate is th e first coordinate in which their associated ft- tuples differ (otherwise, there are no edges between Fi8 an d Ftί ). I t is th en seen t h a t K2kr = U iU <?,. F o r each i —1, 2, , ft, define F<,y t o be t h e set of all vertices v such t h a t v is a vertex of an Fu whose associated ft- tuple (clf c2, , ck ) h as ct = i ; i = 1, 2, • • • , ^ - 1. Then {V<fl, F i> 2 , • • • , F ^ ^ . J is a partition of F ( G J for which t h e subgraph (Vij) consists of rftibi — 1) pairwise disjoint copies of Ft9 j = 1, 2, , ^ —1. Thus, < F i f J > is an acyclic graph for each such j . H ence, α(G {) ^ nt — 1, i = Ί f 2 f . . . , ft. REF EREN CES 1. L. W. Beineke, Decompositions of complete graphs into forests, Magyar Tud. Akad. M at. Kutato I n t . KozL, 9 (1964), 589- 594. 2. J . E. G raver and J . Yackel, Some graph theoretic results associated with Ramsey's theorem, J . Combinatorial Theory, 4 (1968), 125- 175. 3. R. E. G reenwood and A. M. G leason, Combinatorial relations and chromatic graphs, Canad. J . M ath., 7 (1955), 1- 7. 4. C. St. J . A. N ash- Williams, Edge- disjoint spanning trees of finite graphs, J . London M ath. So c , 3 6 (1961), 445- 450. 5. F . P . Ramsey, On a problem of formal logic, Proc. London Math. Soc, 3 0 (1930), 264- 286. RAMSEY THEORY AND CHROMATIC NUMBERS 43 Received October 4, 1974. The second author's research was supported by the Research Foundation of the State of New York. WESTERN MICHIGAN UNIVERSITY AND SUNY, COLLEGE AT FREDONIA PACIFIC JOURNAL OF MATHEMATICS ED ITORS RICHARD AR E N S (M an agin g E ditor) U niversity of California Los Angeles, California 90024 J . DUGUNDJI D epartment of Mathematics U niversity of Southern California Los Angeles, California 90007 R. D . GlLBARG AND J. MlLGRAM A. BE AU M O N T U niversity of Washington Seattle, Washington 98105 Stanford U niversity Stanford, California 94305 ASSOCIATE ED ITORS E. F. BECKENBACH B. H . NEUMANN F . WOLF K . YOSHIDA SU PPORTIN G IN STITU TION S UNIVERSITY OF BRITISH COLUMBIA CALIFORN IA IN STITU TE OF TECHNOLOGY UN IVERSITY OF CALIFORN IA MONTANA STATE U N IVERSITY UN IVERSITY OF NEVADA NEW MEXICO STATE U N IVERSITY OREGON STATE U N IVERSITY UN IVERSITY OF OREGON OSAKA U N IVERSITY U N IVERSITY OF SOUTHERN CALIFORN IA STAN FORD UN IVERSITY U N IVERSITY OF TOKYO U N IVERSITY OF UTAH WASHING TON STATE UN IVERSITY U N IVERSITY OF WASHINGTON * * * AMERICAN MATHEMATICALzyxwvutsrqponmlkjih SOCIETY NAVAL WEAPONS CENTER Printed in Japan by ίntarnational Academic P rinting Co., Ltd., Tokyo, Japan Pacific Journal of Mathematics Vol. 55, No. 1 September, 1974 Robert Lee Anderson, Continuous spectra of a singular symmetric differential operator on a Hilbert space of vector-valued functions . . . . Michael James Cambern, The isometries of L p (X, K ) . . . . . . . . . . . . . . . . . . . R. H. 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Stephen Michael Gagola, Jr., Characters fully ramified over a normal subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frank Larkin Gilfeather, Operator valued roots of abelian analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. S. Goel, A. S. B. Holland, Cyril Nasim and B. N. Sahney, Best approximation by a saturation class of polynomial operators . . . . . . . . . James Secord Howland, Puiseux series for resonances at an embedded eigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . David Jacobson, Linear GCD equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. H. 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Kunio Yamagata, The exchange property and direct sums of indecomposable injective modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 9 19 39 45 55 65 85 93 107 127 149 157 177 195 207 219 233 249 277 285 289 301