Multicolor bipartite Ramsey numbers for quadrilaterals and stars

Xuemei Zhang, Chunyan Weng and Yaojun Chen

Applied Mathematics and Computation, 2023, vol. 438, issue C

Abstract: For bipartite graphs H1,…,Hμ, μ≥2, the μ-color bipartite Ramsey number, denoted by Rb(H1,…,Hμ), is the least positive integer N such that if we arbitrarily color the edges of a complete bipartite graph KN,N with μ colors, then it contains a monochromatic copy of Hi in color i for some i, 1≤i≤μ. Let C4 and K1,n be a quadrilateral and a star on n+1 vertices, respectively. In this paper, we show that the (μ+1)-color bipartite Ramsey number Rb(C4,…,C4,K1,n)≤n+⌈12μ2(4n+μ2+2μ−7)+4⌉+μ2+μ2−1. Moreover, using algebraic methods, we construct Ramsey graphs or near Ramsey graphs and determine infinitely many values of Rb(C4,…,C4,K1,n), which reach the upper bound if μ=1,2 and are at most ⌊μ2⌋ less than the upper bound if μ≥3.

Keywords: Quadrilateral; Star; Galois field; Difference set; Multicolor bipartite Ramsey number (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:438:y:2023:i:c:s0096300322006506

DOI: 10.1016/j.amc.2022.127576

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