Abstract
Let k be a positive integer and let F and \(H_{1}, H_{2}, \ldots , H_{k}\) be simple graphs. The proper-Ramsey number \(pr_{k}(F; H_{1}, H_{2}, \ldots , H_{k})\) is the minimum integer n such that any k-coloring of the edges of \(K_{n}\) contains either a properly colored copy of F or a copy of \(H_{i}\) in color i, for some i. We consider the case where \(F = C_{4}\) is fixed, and establish the exact value of the proper-Ramsey number when \(\{H_i\}_{i=1}^k\) is a family containing only cliques, and nearly sharp bounds for the proper-Ramsey number when \(\{H_i\}_{i=1}^k\) is a family containing only cycles or only stars. We also give a general bound for the proper-Ramsey number that is nearly tight when \(\{H_i\}_{i=1}^k\) is a family of maximal split graphs.
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A graph in which the vertices can be partitioned into a clique and an independent set.
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Daniel M. Martin: Partially funded by CNPq proc. 311789/2015-3.
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Magnant, C., Martin, D.M. & Salehi Nowbandegani, P. Monochromatic Subgraphs in the Absence of a Properly Colored 4-Cycle. Graphs and Combinatorics 34, 1147–1158 (2018). https://doi.org/10.1007/s00373-018-1955-z
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DOI: https://doi.org/10.1007/s00373-018-1955-z