Size multipartite Ramsey numbers for stripes versus small cycles

Abstract

For simple graphs G 1 and G 2 , the size Ramsey multipartite number m j (G 1 , G 2 ) is defined as the smallest natural number s such that any arbitrary two coloring of the graph K j×s using the colors red and blue, contains a red G 1 or a blue G 2 as subgraphs. In this paper, we obtain the exact values of the size Ramsey numbers m j (nK 2 , C m ) for j ≥ 2 and m ∈ {3, 4, 5, 6}.

The Size, Multipartite Ramsey Numbers for nK2 Versus Path–Path and Cycle

“…In [10], Jayawardene et al determined the size of the multipartite Ramsey numbers m j (nK 2 , C m ) where j ≥ 2 and m ∈ {3, 4, 5, 6}. The second goal of this work extends these results, as stated below.…”

“…For j = 2, since the bipartite graph has no odd cycle, we have m 2 (nK 2 , C 7 ) = ∞. For other cases, we start with the following proposition: [10], and 3K 2 ⊆ G, we have…”

“…The existence of such a positive integer is guaranteed by a result in [2]. The size of the multipartite Ramsey numbers of small paths versus certain classes of graphs have been studied in [8][9][10]. The size of the multipartite Ramsey numbers of stars versus certain classes of graphs have been studied in [11,12].…”

“…In [13,14], Burger, Stipp, Vuuren, and Grobler investigated the multipartite Ramsey numbers m j (G 1 , G 2 ), where G 1 and G 2 are in a completely balanced multipartite graph, which can be naturally extended to several colors. Recently, the numbers m j (G 1 , G 2 ) have been investigated for special classes: stripes versus cycles; and stars versus cycles, see [10] and its references. In [15], authors determined the necessary and sufficient conditions for the existence of multipartite Ramsey numbers m j (G, H) where both G and H are incomplete graphs, which also determined the exact values of the size multipartite Ramsey numbers m j (K 1,m , K 1,n ) for all integers m, n ≥ 1 and j = 2, 3.…”

Three-colour bipartite Ramsey number $R_b(G_1,G_2,P_3)$

“…Some variants of graph Ramsey number appear as a result of generalisation of the host graphs K n and K n,n , where complete balanced multipartite graph takes the place. For example, set multipartite Ramsey number [3] and size multipartite Ramsey number [4,13,14,16].…”