The m-bipartite Ramsey number BR_m(H₁,H₂)

2022/02/10 by Yaser Rowshan, Rowshan, Yaser
Mathematics · Computer Science · #Limits and Structures in Graph Theory #Advanced Graph Theory Research #Complexity and Algorithms in Graphs

paper · pdf · doi:10.48550/arxiv.2202.04921

Abstract

In a (G¹,G²) coloring of a graph G, every edge of G is in G¹ or G². For two bipartite graphs H₁ and H₂, the bipartite Ramsey number BR(H₁, H₂) is the least integer b≥ 1, such that for every (G¹, G²) coloring of the complete bipartite graph K_b,b, results in either H₁⊆ G¹ or H₂⊆ G². As another view, for bipartite graphs H₁ and H₂ and a positive integer m, the m-bipartite Ramsey number BR_m(H₁, H₂) of H₁ and H₂ is the least integer n, such that every subgraph G of K_m,n results in H₁⊆ G or H₂⊆ G. The size of m-bipartite Ramsey number BR_m(K_2,2, K_2,2), the size of m-bipartite Ramsey number BR_m(K_2,2, K_3,3) and the size of m-bipartite Ramsey number BR_m(K_3,3, K_3,3) have been computed in several articles up to now. In this paper we determine the exact value of BR_m(K_2,2, K_4,4) for each m≥ 2.

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