Let G be a bipartite graph, with k | e(G). The zero-sum bipartite Ramsey number B(G,Zk) is the smallest integer t such that in every Zk-coloring of the edges of Kt,t, there is a zero-sum mod k copy of G in Kt,t. In this paper we give the first proof which determines B(G,Z2) for all possible bipartite graphs G. In fact, we prove a much more general result from which B(G,Z2) can be deduced: Let G...

A graph $G$ is Ramsey for a $H$ if every colouring of the edges in two colours contains monochromatic copy $H$. Two graphs $H_1$ and $H_2$ are equivalent any only it $H_2$. parameter $s$ distinguishing $s(H_1)\neq s(H_2)$ implies that not equivalent. In this paper we show chromatic number parameter. We also extend to multi-colour case use similar idea find another which distinguishing.

The degree Ramsey number of a graph G, denoted R∆(G; s), is min{∆(H) : H s → G}, where H s → G means that every s-edge-coloring of H contains a monochromatic copy of G. The closed k-blowup of a graph is obtained by replacing every vertex with a clique of size k and every edge with a complete bipartite graph where both partite sets have size k. We prove that there is a function f such that R∆(G;...

We determine the maximum number of colors in a coloring of the edges of Km;n such that every cycle of length 2k contains at least two edges of the same color. One of our main tools is a result on generalized path covers in balanced bipartite graphs. For positive integers q a, let g(a;q) be the maximum number of edges in a spanning subgraph G of Ka;a such that the minimum number of vertex-disjoi...

The size Ramsey number of a graph H is defined as the minimum edges in G such that there monochromatic copy every two-coloring E(G). was introduced by Erdős, Faudree, Rousseau, and Schelp 1978 they ended their foundational paper asking whether one can determine up to constant factor numbers three families graphs: complete bipartite graphs, book graphs (obtained adding many common neighbors vert...

Motivated by questions in Ramsey theory, Thomason and Wagner described the edge colorings of complete graphs that contain no rainbow path Pt of order t. In this paper, we consider the edge colorings of complete bipartite graphs that contain no rainbow path Pt. Mathematics Subject Classification: 05C15, 05C38, 05C55