Abstract
Given two graphs G and H, the Ramsey number R(G,H) is the minimum integer N such that any two-coloring of the edges of KN in red or blue yields a red G or a blue H. Let v(G) be the number of vertices of G and χ(G) be the chromatic number of G. Let s(G) denote the chromatic surplus of G, the number of vertices in a minimum color class among all proper χ(G)-colorings of G. Burr showed that \(R(G,H) \ge (v(G) - 1)(\chi (H) - 1) + s(H)\) if G is connected and \(v(G) \ge s(H)\). A connected graph G is H-good if \(R(G,H) = (v(G) - 1)(\chi (H) - 1) + s(H)\). Let tH denote the disjoint union of t copies of graph H, and let \(G \vee H\) denote the join of G and H. Denote a complete graph on n vertices by Kn, and a tree on n vertices by Tn. Denote a book with n pages by Bn, i.e., the join \({K_2} \vee \overline {{K_n}} \). Erdős, Faudree, Rousseau and Schelp proved that Tn is Bm-good if \(n \ge 3m - 3\). In this paper, we obtain the exact Ramsey number of Tn versus 2B2- Our result implies that Tn is 2B2-good if n ≥ 5.
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This paper is supported in part by National Natural Science Foundation of China (No. 11931002) and China Postdoctoral Science Foundation (No. 2021M701162).
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Guo, Xb., Hu, Sn. & Peng, Yj. Ramsey Numbers of Trees Versus Multiple Copies of Books. Acta Math. Appl. Sin. Engl. Ser. (2023). https://doi.org/10.1007/s10255-024-1117-4
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DOI: https://doi.org/10.1007/s10255-024-1117-4