Some Bistar Bipartite Ramsey Numbers |
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Authors: | Johannes H Hattingh Ernst J Joubert |
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Institution: | 1. Department of Mathematics, East Carolina University, Greenville, NC, 27858, USA 2. Department of Mathematics, University of Johannesburg, Auckland Park, South Africa
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Abstract: | For bipartite graphs G 1, G 2, . . . ,G k , the bipartite Ramsey number b(G 1, G 2, . . . , 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. A tree of diameter three is called a bistar, and will be denoted by B(s, t), where s ≥ 2 and t ≥ 2 are the degrees of the two support vertices. In this paper we will obtain some exact values for b(B(s, t), B(s, t)) and b(B(s, s), B(s, s)). Furtermore, we will show that if k colours are used, with k ≥ 2 and s ≥ 2, then \({b_{k}(B(s, s)) \leq \lceil k(s - 1) + \sqrt{(s - 1)^{2}(k^{2} - k) - k(2s - 4)} \rceil}\) . Finally, we show that for s ≥ 3 and k ≥ 2, the Ramsey number \({r_{k}(B(s, s)) \leq \lceil 2k(s - 1)+ \frac{1}{2} + \frac{1}{2} \sqrt{(4k(s - 1) + 1)^{2} - 8k(2s^{2} - s - 2)} \rceil}\) . |
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