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Some Ramsey Numbers for Complete Bipartite Graphs

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Some Ramsey Numbers for Complete Bipartite Graphs Some Ramsey Numbers for Complete Bipartite Graphs Martin Harborth Otto-von-Guericke-UniversiUit Magdeburg Magdeburg~ Germany Ingrid Mengersen Technische Universitat Braunschweig Braunschweig, Germany Abstract Upper bounds are determined for the Ramsey number ](m,n), 2 :; m n. These bounds are attained for infinitely many n in case of m 3 and are fairly close to the exact value for every m if n is sufficiently large. 1 Introduction For complete bipartite graphs G and H only few exact values of the Ramsey number r(G, H) are known. determined the numbers r(I<l,t, I{I,n)' .Parsons [7,8,9] and Stevens [10] investigated the numbers r(I<I,t,I{m,n)' Parsons determined an upper bound for the case m = 2, which is attained if n is small relative to t and certain regular graphs exist. Stevens completely solved the case when n is sufficiently depending on t and m. In [3] it was shown that r(I<2,n, K 2,n) 4n - 2 with equality for infinitely many n. Moreover, Chung and Graham [1] derived a general upper bound for r(Km,n, But up to now, besides r(I(~,3' K3,3) 18 determined in [3], no exact values of r(1(,.t, Km,n) are known when s, t, m, n ~ 3. Here we focus on the numbers r(K2,2, Km,n) = r(04' Km,n)' The case m = 1 was already studied by Parsons. He showed that r( C4 , KI,n) :::;: n + rfol + 1 with equality for infinitely many n. Here we will derive corresponding upper bounds for the case m 2: 2. These bounds are attained for infinitely many n in case of m :::;: 3 and are fairly close to the exact value for fixed m and sufficiently large n. As usual, the vertex set of a graph G is denoted by V and the edge set by E. Na( v) denotes the set of neighbors of a vertex v E G in G and da( v) the degree of v in G. The minimum degree of the vertices in G is denoted by oa and the maximum degree by ~a. In a 2-coloring of the complete graph Kn we always use green and red as colors. The green sub graph is denoted by G(g) and the red subgraph by G(r). We write Ng(v), dg(v), Og and ~g instead of NG(g)(v), da(g)(v), oG(g) and ~a(g) and use the corresponding notations for G(r). If A and B are two sets of vertices from K n , g(A, B) denotes the number of green edges from A to B. If A consists of a single vertex u, we write g( u, B). A 2-coloring of Kn is said to be a (G, H)-coloring if there is neither a green subgraph G nor a red subgraph H. Australasian Journal of Combinatorics 13(1996). pp.119-128 2 Properties of (C4 ,Km ,n)-colorings The following lemmas will be used later to establish upper bounds for r(C41 Km,n)' Lemma 1. Let X be a (C4 , Km,n)-coloring of Kp and v E V. Then the following assertions hold. (i) L g(v" Nr(v)) S p - dg(v) - L (1) uENg(v) (ii) If dg(v) 2: m and if 8 is an m-element subset ofN.q(v) then (2) uES uES (iii) If dg ( v) ~ m - 1 and if S' is an (m - }-eleIUeIH subset of Ng(v) then Lg(u, Nr(v)) :2 p - n - dg(v). (3) uES Proof. Since no green C 4 occurs in X, each vertex in N r ( v) can be joined by at most one green edge to Ng(v) and this yields (1). Moreover, there is no red Km,n in X. in Ng(v) there are no m vertices with n common red in Nr(v)UNg(v) and no m - 1 with n common red neighbors in Nr(v). This implies (2) and (3). II Lemma 2. Let X be a Km,n)-coloring of KPl m ~ 2 and p ~ max{ n + m + 2 1, n + m - m - I}. Then p - n - 11 p - n J r m + 1 S; 6. g S; m + l(m + n - 1) / U--;;;-1 - 1) . (4) Furthermore, p-n dg(v) J dg ( v) S; m - 1 + (n - 1)/ r 1 (5) l m -1 for every v E V with m 1 S; dg ( v) S p - n - 1. Proof. No red Km,n implies green edges in X. Take m vertices at least two of them adjacent in green. They have at least p-m m6.g +2 and at most n 1 common red neighbors. This implies 6. g 2: m - 1. Consider now a vertex v with dg(v) = 6.g • Let lV.q( v) = {Ul' ... , U6. g } and gi = g( Ui, Nr ( v)). We may assume that gl S; g2 S; ... S g6. g Using inequality (3) and gi S; 6. g - 1 we obtain that (m - 1)(6.9 1) ~ 2::::~1 gi :2 p - n - 6. g • This yields the first inequality in (4). To prove the second inequality in (4) note that 6.g ~ m. Moreover, In::l Nr(Ui) n Ng(v)1 ~ 6.g 2m since each 11i can have at most one green neighbor in Ng ( v). Now inequality (2) implies that 2:::1 gi 2: p - n - 2m which yields gm 2: r(2::7~1 gi)/m1 2: np - n)/m1 - 2. Using 120 that gm :::; gmH :::; ... :::; and inequality (1) we obtain that p n 2m + (D. g m) ( r(p n) / m 1 2):::; g, :::; p -- 1 and this yields the second inequality of (4). To prove inequality (.5) consider a vertex v with d = dg(v) .2: m 1 and d :::; p-n-l. Let Ng(v) {uh ... ~ud}andgi g(ui,Nr(v)). Again we may assume that gl :::; ... :::; gd· Then inequalities (3) and (1) imply that p n - d + (d (m 1) H(p- n d)j(m l)l p d 1 yielding inequality (5). II1II 3 Erdos-Renyi and Moore graphs Here we consider two classes of graphs which will be useful to establish lower bounds for r( C4~ For a prime power q the Erdos-Renyi graph E R( q), first constructed by Erdos and Renyi in [2], defined to be the graph whose vertices are the points of the projective plane PG(2, q) where two vertices y, z) and ~ y', are adjacent iff xx' + + zz' O. The Erdos-Renyi was studied in detail Parsons in [9]. Here we will use the following properties of E R( q). (0:) ER(q) has q2 + q+ 1 "fl"""~~~ ((3) ER(q) does not contain subgraph (,) In ER(q) there are no two vertices of degree q. (8) In E R( q) no vertex of degree q belongs to a subgraph Lemma 3. Let q be a prime power; G ER(q), G the complement of G and let {VI. V m } C V V(G). Then (6) Proof. Let T V\S and, for v E let Tv N G ( v) n T and NG(v) n S. Then I Nc(v)1 -I UVES 7~1 and, by property (0:), ITI q2 + q + 1 - m. Thus, inequality (6) is proved if we can show that IU .2:qm (m2')' (7) vES Let M {(i, j); 1 S; i < j m}. Trivially, ITvl - 2::= ITvi n Tv} I· (8) (i,j)EM Let Ml {(i,j) E M;{Vi,Vj} E E(G), min{dG(vi),dG(v;)} q} and }\l12 = {(i,j) E M; ISVi n Svjl I}. By properties ((3) and (8), (;) -IN111-INI21· (9) (i,j)EM 121 Let S' {v E S; dG(v) q} and S" = S\S'. By property (,), IMII = LVES1 ISvl. Furthermore, by property (13). IM21 = ZVES (ISvl) Thus, inequalities and (9) imply Note that dG(v) and, by property (a), the vertices in S" have degree q + 1. Thus, every summand of the two sums in (10) at least q. This proves (7) and the proof of Lemma 3 is complete. III For integers /j 3 and 9 2: 3 a (/j, g)- Moore graph is defined to be a regular of /j with girth 9 and p(/j, g) vertices where _ { 1 {( 8 - 1 - I} if 9 is odd + 5~Z (11 ) p((5,g) - 5~2{((5 _1)g/2 I} if 9 IS even. It is well kno'wn that every graph with minimum 8 and 9 has at least p( /j, g) vertices. In the following section we will use a result of Hoffman and Singleton [6] concerning (/j,5)-Moore graphs. They showed that there are no such graphs with (5 2: 3 and (5 rf. {3, 7, 57} whereas (3,5)- and (7,5)-Moore do exist (the Petersen graph and the so-called Hoffman-Singleton graph). Up to now it is unknown whether a (57, 5)-Moore graph exists. 4 Ramsey numbers r(C4 , Km,n) We will determine bounds and values for r( C4 , I<m,n) which depend in case of 2 :S m :S 4 on the difference s between nand (ffol - 1)2, the square less than n (1 :S .5 2 - 1). Theorem 1. Let n 2: 2, q = hlnl, s = n - (q - 1)2 and j\ll = {2, 5, :37, :n37}. Then (12) Proof. Suppose first that we have a (C4 , I<z,n)- coloring of I<p where p = n + 2q and 1 :S s :S q - 1.
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