Article

# On the Ramsey Problem for Multicolor Bipartite Graphs

Department of Mathematics, State University of Maringá, Maringá, PR, Brazil
(Impact Factor: 0.82). 01/1999; 22(1):48-59. DOI: 10.1006/aama.1998.0620

ABSTRACT

Giveni, jpositive integers, letKi, jdenote a bipartite complete graph and letRr(m, n) be the smallest integerasuch that for anyr-coloring of the edges ofKa, aone can always find a monochromatic subgraph isomorphic toKm, n. In other words, ifa ≥ Rr(m, n) then every matrixa × awith entries in {0, 1,…,r − 1} always contains a submatrixm × norn × mwhose entries arei, 0 ≤ i ≤ r − 1. We shall prove thatR2(m, n) ≤ 2m(n − 1) + 2m − 1 − 1, which generalizes the previous resultsR2(2, n) ≤ 4n − 3 andR2(3, n) ≤ 8n − 5 due to Beineke and Schwenk. Moreover, we find a class of lower bounds based on properties of orthogonal Latin squares which establishes that limr → ∞ Rr(2, 2)r − 2 = 1.

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• "In a more recent article [3], Carnielli and Carmelo showed that in any 2-coloring of a bipartite complete graph K n,n , one can always find a monochromatic subgraph isomorphic to K p,q if n ≥ 2 p (q − 1) + 2 p−1 − 1. As in asymptotic versions [4] [9], Caro and Rousseau achieved that there are constants c 1 and c 2 such that "
##### Article: On the Ramsey Numbers for Bipartite Multigraphs
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ABSTRACT: A coloring of a complete bipartite graph is shuffle-preserved if it is the case that assigning a color $c$ to edges $(u, v)$ and $(u', v')$ enforces the same color assignment for edges $(u, v')$ and $(u',v)$. (In words, the induced subgraph with respect to color $c$ is complete.) In this paper, we investigate a variant of the Ramsey problem for the class of complete bipartite multigraphs. (By a multigraph we mean a graph in which multiple edges, but no loops, are allowed.) Unlike the conventional m-coloring scheme in Ramsey theory which imposes a constraint (i.e., $m$) on the total number of colors allowed in a graph, we introduce a relaxed version called m-local coloring which only requires that, for every vertex $v$, the number of colors associated with $v$'s incident edges is bounded by $m$. Note that the number of colors found in a graph under $m$-local coloring may exceed m. We prove that given any $n \times n$ complete bipartite multigraph $G$, every shuffle-preserved $m$-local coloring displays a monochromatic copy of $K_{p,p}$ provided that $2(p-1)(m-1) < n$. Moreover, the above bound is tight when (i) $m=2$, or (ii) $n=2^k$ and $m=3\cdot 2^{k-2}$ for every integer $k\geq 2$. As for the lower bound of $p$, we show that the existence of a monochromatic $K_{p,p}$ is not guaranteed if $p> \lceil \frac{n}{m} \rceil$. Finally, we give a generalization for $k$-partite graphs and a method applicable to general graphs. Many conclusions found in $m$-local coloring can be inferred to similar results of $m$-coloring.
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• "In the third section we consider the diagonal numbers b(K 2;n ; K 2;n ). Several papers, for example [1] [5] [10] [14], have considered distinct approaches to numbers b(G 1 ; G 2 ), either studying the connections to the Zarankiewicz problem, or investigating connections to the Hadamard matrices. Beineke and Schwenk [1] proved that if there exists a Hadamard matrix of order 2(n − 1), with odd n, then b(K 2;n ; K 2;n ) = 4n − 3. Therefore, the values of b(K 2;n ; K 2;n ) are known when n is a small odd number. "
##### Article: K_{2,2}-K_{1,n} and K_{2,n}-K_{2,n} bipartite Ramsey numbers
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ABSTRACT: Some bounds for G 1 --G 2 bipartite Ramsey numbers b(G 1 ; G 2 ) are given, which imply that b(K 2; 2 ; K 1;n )=n + q for the range q 2 - q +16n6q 2 , where q is a prime power. Our new construction establishes in particular that b(K 2;n ; K 2;n )=4n - 3if4n - 3 is a prime power, reinforcing a weaker form of a conjecture due to Beineke and Schwenk. Particular relationships between b(G 1 ; G 2 ) and G 1 --G 2 Ramsey numbers are also determined. c # 2000 Elsevier Science B.V. All rights reserved. MSC: 05C55 Keywords: Generalized Ramsey theory; Bipartite Ramsey number; Strongly regular graphs 1. Introduction Given G 1 and G 2 bipartite graphs, the bipartite Ramsey number b(G 1 ; G 2 )isthe least positive integer b such that if the edges of K b; b are colored with two colors, then there will always exist an isomorphic copy of G 1 in the #rst color, or a copy of G 2 in the second color. In other words, b(G 1 ; G 2 ) is the least positive integer b such that given any subgraph H...
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• "In the third section we consider the diagonal numbers b(K 2;n ; K 2;n ). Several papers, for example [1] [5] [10] [14], have considered distinct approaches to numbers b(G 1 ; G 2 ), either studying the connections to the Zarankiewicz problem, or investigating connections to the Hadamard matrices. Beineke and Schwenk [1] proved that if there exists a Hadamard matrix of order 2(n − 1), with odd n, then b(K 2;n ; K 2;n ) = 4n − 3. Therefore, the values of b(K 2;n ; K 2;n ) are known when n is a small odd number. "
##### Article: K2,2-K1,n and K2,n-K2,n bipartite Ramsey numbers
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ABSTRACT: Some bounds for G 1 -G 2 bipartite Ramsey numbers b(G 1 ;G 2 ) are given, which imply that b(K 2,2 ;K 1,n )=n+q for the range q 2 -q+1≤n≤q 2 , where q is a prime power. Our new construction establishes in particular that b(K 2,n ;K 2,n )=4n-3 if 4n-3 is a prime power, reinforcing a weaker form of a conjecture due to Beineke and Schwenk. Particular relationships between b(G 1 ;G 2 ) and G 1 -G 2 Ramsey numbers are also determined.
Discrete Mathematics 08/2000; 223(1). DOI:10.1016/S0012-365X(00)00041-8 · 0.56 Impact Factor