On size multipartite Ramsey numbers for paths versus stars (original) (raw)
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On size multipartite Ramsey numbers for stars versus paths and cycles
Electronic Journal of Graph Theory and Applications
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 1 ⊕ F 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.
Star-path bipartite Ramsey numbers
Discrete Mathematics, 1998
For bipartite graphs G1,G2 ..... Gk, the bipartite Ramsey number b(GI,G2,...,Gk) is the least positive integer b so that any colouring of the edges of Kb, b with k colours will result in a copy of Gi in the ith colour for some i. In this note, we establish the exact value of the bipartite Ramsey number b(Pm,Kl,,) for all integers re, n>.2, where Pm denotes a path on m vertices.
On size multipartite Ramsey numbers for stars
Indonesian Journal of Combinatorics
Burger and Vuuren defined the size multipartite Ramsey number for a pair of complete, balanced, multipartite graphs mj(Kaxb,Kcxd), for natural numbers a,b,c,d and j, where a,c >= 2, in 2004. They have also determined the necessary and sufficient conditions for the existence of size multipartite Ramsey numbers mj(Kaxb,Kcxd). Syafrizal et al. generalized this definition by removing the completeness requirement. For simple graphs G and H, they defined the size multipartite Ramsey number mj(G,H) as the smallest natural number t such that any red-blue coloring on the edges of Kjxt contains a red G or a blue H as a subgraph. In this paper, we determine the necessary and sufficient conditions for the existence of multipartite Ramsey numbers mj(G,H), where both G and H are non complete graphs. Furthermore, we determine the exact values of the size multipartite Ramsey numbers mj(K1,m, K1,n) for all integers m,n >= 1 and j = 2,3, where K1,m is a star of order m+1. In addition, we also d...
Size multipartite Ramsey numbers for small paths versus books
Indonesian Journal of Combinatorics
Given jge2j \ge 2jge2, for graphs GGG and HHH, the size Ramsey multipartite number mj(G,H)m_j(G, H)mj(G,H) is defined as the smallest natural number ttt such that any blue red coloring of the edges of the graph KjtimestK_{j \times t}Kjtimest, necessarily containes a red GGG or a blue HHH as subgraphs. Let the book with nnn pages is defined as the graph K1+K1,nK_1 + K_{1,n}K1+K1,n and denoted by BnB_nBn. In this paper, we obtain the exact values of the size Ramsey numbers mj(P3,H)m_j(P_3, H)mj(P3,H) for jge3j \ge 3jge3 where HHH is a book BnB_nBn. We also derive some upper and lower bounds for the size Ramsey numbers mj(P4,H)m_j(P_4, H)mj(P4,H) where HHH is a book BnB_nBn.
The Size, Multipartite Ramsey Numbers for nK2 Versus Path–Path and Cycle
Mathematics
For given graphs G1,G2,…,Gn and any integer j, the size of the multipartite Ramsey number mj(G1,G2,…,Gn) is the smallest positive integer t such that any n-coloring of the edges of Kj×t contains a monochromatic copy of Gi in color i for some i, 1≤i≤n, where Kj×t denotes the complete multipartite graph having j classes with t vertices per each class. In this paper, we computed the size of the multipartite Ramsey numbers mj(K1,2,P4,nK2) for any j,n≥2 and mj(nK2,C7), for any j≤4 and n≥2.
Graphs and Combinatorics, 2015
For two given graphs G 1 and G 2 , the Ramsey number R(G 1 , G 2) is the least integer r such that for every graph G on r vertices, either G contains a G 1 or G contains a G 2. In this note, we determined the Ramsey number R(K 1,n , W m) for even m with n + 2 ≤ m ≤ 2n − 2, where W m is the wheel on m + 1 vertices, i.e., the graph obtained from a cycle C m by adding a vertex v adjacent to all vertices of the C m .
On multicolour noncomplete Ramsey graphs of star graphs
Discrete Applied Mathematics, 2008
Given graphs G, G 1 ,. .. , G k , where k ≥ 2, the notation G → (G 1 , G 2 ,. .. , G k) denotes that every factorization F 1 ⊕ F 2 ⊕ • • • ⊕ F k of G implies G i ⊆ F i for at least one i, 1 ≤ i ≤ k. We characterize G for which G → (K(1, n 1), K(1, n 2),. .. , K(1, n k)) and derive some consequences from this. In particular, this gives the value of the graph Ramsey number R(K(1, n 1), K(1, n 2),. .. , K(1, n k)).
On Ramsey theory and graphical parameters
Pacific Journal of Mathematics, 1977
A graph G is said to have a factorization into the subgraphs G u-, G k if the subgraphs are spanning, pairwise edge-disjoint, and the union of their edge sets equals the edge set of G. For a graphical parameter / and positive integers rti, n 2 , , n k (k ^ 1), the /-Ramsey number f{n u n 2 , , n k) is the least positive integer p such that for any factorization K p = Uΐ=iG l9 it follows that /(G)^n t for at least one i, l^i^k. In the following, we present two results involving /-Ramsey numbers which hold for various vertex and edge partition parameters, respectively. It is then shown that the concept of /-Ramsey number can be generalized to more than one vertex partition parameter, more than one edge partition parameter, and combinations of vertex and edge partition parameters. Formulas are presented for these generalized /-Ramsey numbers and specific illustrations are given.
2001
For a graph G, a partiteness k ≥ 2 and a number of colours c, we define the multipartite Ramsey number r c k (G) as the minimum value m such that, given any colouring using c colours of the edges of the complete balanced k-partite graph with m vertices in each partite set, there must exist a monochromatic copy of G. We show that the question of the existence of r c k (G) is tied up with what monochromatic subgraphs are forced in a ccolouring of the complete graph K k . We then calculate the values for some small G including r 2 3 (C 4 ) = 3, r 2 4 (C 4 ) = 2, r 3 3 (C 4 ) = 7 and r 2 3 (C 6 ) = 3.