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Some Generalized Bipartite Ramsey Numbers Involving Short Cycles

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Abstract

For bipartite graphs \(G_1, G_2,\ldots ,G_k\), the bipartite Ramsey number \(b(G_1,G_2\), \(\ldots , G_k)\) is the least positive integer b so that any colouring of the edges of \(K_{b,b}\) with k colours will result in a copy of \(G_i\) in the ith colour for some i. In this paper, we will consider the bipartite Ramsey number \(b(C_{2t_1},C_{2t_2},\ldots ,C_{2t_k})\), where \(t_{i}\) is an integer and \(2 \le t_{i}\le 4,\) for all \(1\le i\le k\). In particular, we will show that \(b(C_{2t_1},C_{2t_2},\ldots ,C_{2t_k})\) \(\le \) \(k(t_1+t_2+\cdots +t_k-k+1)\).

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Correspondence to Ernst J. Joubert.

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Joubert, E.J. Some Generalized Bipartite Ramsey Numbers Involving Short Cycles. Graphs and Combinatorics 33, 433–448 (2017). https://doi.org/10.1007/s00373-017-1761-z

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