Abstract
For graphs G and H, the Ramsey number \(r_{k+1}(G;H)\) is defined as the minimum N such that any edge-coloring of \(K_N\) by \(k+1\) colors contains either a monochromatic G in the first k colors or a monochromatic H in the last color. A book \(B^{(m)}_{n}\) is a graph that consists of n copies of \(K_{m+1}\) sharing a common \(K_m\). We shall give upper bounds for \(r_{k+1}(K_{t,s};B^{(m)}_n)\) and \(r_{k+1}(C_{2t};B^{(m)}_n)\), some of which are sharp up to the sub-linear term asymptotically.
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References
Bondy, J., Simonovits, M.: Cycles of even length in graphs. J. Combin. Theory Ser. B 16, 97–105 (1974)
Chung, F., Graham, R.L.: On multicolor Ramsey numbers for complete bipartite graphs. J. Combin. Theory Ser. B 18, 164–169 (1975)
Conlon, D.: The Ramsey number of books. Adv. Combin. 3, 12pp (2019)
Füredi, Z.: An upper bound on Zarankiewicz’ problem. Combin. Probab. Comput. 5, 29–33 (1996)
Kövari, T., Sós, V., Turán, P.: On a problem of K. Zarankiewicz. Colloq. Math. 3, 50–57 (1954)
Li, Y., Rousseau, C.C.: On book-complete graph Ramsey numbers. J. Combin. Theory Ser. B 68, 36–44 (1996)
Lin, Q., Peng, X.: Large book-cycle Ramsey numbers. SIAM J. Discrete Math. 35, 532–545 (2021)
Liu, M., Li, Y.: Ramsey numbers of a fixed odd-cycle and generalized books and fans. Discrete Math. 339, 2481–2489 (2016)
Parsons, T.: Ramsey graphs and block designs. I. Trans. Amer. Math. Soc. 209, 33–44 (1975)
Rousseau, C.C., Sheehan, J.: On Ramsey numbers for books. J. Graph Theory 2, 77–87 (1978)
Wang, Y., Li, Y., Li, Y.: Ramsey numbers of several \(K_{t, s}\) and a large \(K_{m, n}\). Discrete Math. 345, 112987 (2022)
Wenger, R.: Extremal graphs with no \(C_4\)’s, \(C_6\)’s, or \(C_{10}\)’s. J. Combin. Theory Ser. B 52, 113–116 (1991)
Zhang, X., Chen, Y., Cheng, T.: On three color Ramsey numbers \(R(C_4, C_4, K_{1, n})\). Discrete Math. 342, 285–291 (2019)
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We are grateful to the editors and referees for invaluable suggestions and comments that improve the presentation of the paper greatly.
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This work is supported in part by NSFC (11871377,11931002,12101156)
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Li, Y., Li, Y. & Wang, Y. Multicolor Ramsey Numbers of Bipartite Graphs and Large Books. Graphs and Combinatorics 39, 21 (2023). https://doi.org/10.1007/s00373-023-02623-1
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DOI: https://doi.org/10.1007/s00373-023-02623-1