# Lower bounds of the size multipartite Ramsey numbers mj(Pn,KjÃ—b)

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Let F,G, and H be simple graphs. We say F â†’ (G,H) if for every 2-coloring of the edges of F there exist a monochromatic G or H in F. The Ramsey number r(G;H) is defined as min f|V(F)| : F â†’ (G,H)g, the size Ramsey number È“(G,H) is defined as min f|E(F)| : F â†’(G,H), and...

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A two-colored digraph D(2) is a digraph D whose each of its arc is colored by either red or blue. For nonnegative integers s and t with s+t > 0, an (s, t)-walk in a two-colored digraph is a walk of length s+t consisting of s red arcs and t blue arcs. The vector (s, t)T is the composition of the...

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Let F, G, and H be finite, simple, and undirected graphs. The notation F â†’ (G,H) means that if the edge set of F is arbitrarily colored by red or blue, then there always exists either a red copy of G or a blue copy of H. The connected size Ramsey number È“c(G,H) is min{|E(F)| : F...

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For graphs G1 and G2, the size bipartite Ramsey number m2(G1, G2 ) is the least natural number t such that any red-blue coloring on the edges of K jxt necessarily forces a red G or a blue H as a subgraph. In this note, we determine the exact values of m2(Pn, Ts ) for all integers n, s â‰¥2,...

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For given graphs G and G, the Ramsey number R( G, G) is the least integer n such that every 2-coloring of the edges of K contains a subgraph isomorphic to G in the first color or a subgraph isomorphic to G in the second color. Surahmat et al. proved that the Ramsey number $${R(C_4, W_n) \leq n +...

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Given two graphs $$G_1$$ and $$G_2$$ , the Ramsey number $$R(G_1, G_2)$$ is the smallest integer $$N$$ such that, for any graph $$G$$ of order $$N$$ , either $$G_1$$ is a subgraph of $$G$$ , or $$G_2$$ is a subgraph of the complement of $$G$$ . We consider the case that $$G_1$$ is a cycle and...

- Split Permutation Graphs. Korpelainen, Nicholas; Lozin, Vadim; Mayhill, Colin // Graphs & Combinatorics;May2014, Vol. 30 Issue 3, p633
The class of split permutation graphs is the intersection of two important classes, the split graphs and permutation graphs. It also contains an important subclass, the threshold graphs. The class of threshold graphs enjoys many nice properties. In particular, these graphs have bounded...