On size multipartite Ramsey numbers for stars versus paths and cycles - Anie Lusiani; Edy Tri Baskoro; Suhadi Wido Saputro (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2017)
Let K_l× t be a complete, balanced, multipartite graph consisting of l partite sets and t vertices in each partite set. For given two graphs G_1 and G_2, and integer j≥ 2, the size multipartite Ramsey number m_j(G_1,G_2) is the smallest integer t such that every factorization of the graph K_j× t:=F_1F_2 satisfies the following condition: either F_1 contains G_1 or F_2 contains G_2. In 2007, Syafrizal et al. determined the size multipartite Ramsey numbers of paths P_n versus stars, for n=2,3 only. Furthermore, Surahmat et al. (2014) gave the size tripartite Ramsey numbers of paths P_n versus stars, for n=3,4,5,6. In this paper, we investigate the size tripartite Ramsey numbers of paths P_n versus stars, with all n≥ 2. Our results complete the previous results given by Syafrizal et al. and Surahmat et al. We also determine the size bipartite Ramsey numbers m_2(K_1,m,C_n) of stars versus cycles, for n≥ 3,m≥ 2.