Ramsey Numbers of Stripes Versus Trees and Unicyclic GraphsRamsey Numbers of Stripes Versus Trees and Unicyclic GraphsS.-N. Hu, Y.-J. Peng

For graphs G and H, the Ramsey number R(G, H) is the minimum integer N such that any coloring of the edges of the complete graph KN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_N$$\end{document} in red or blue yields a red G or a blue H. Denote the union of t disjoint copies of a graph F by tF. We call tK2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$tK_2$$\end{document} a stripe. In this paper, we completely determine Ramsey numbers of stripes versus trees and unicyclic graphs. Our result also implies that a tree is tK2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$tK_2$$\end{document}-good if and only if the independence number of this tree is no less than t. As an application, we improve the known Ramsey numbers of stars versus fan graphs. Moreover, we determine the bipartite Ramsey numbers of a connected bipartite graph versus stripes.