Ramsey and Gallai–Ramsey numbers for comb and sun graphs

Abstract

Given two graphs $G$ and $H$, the Ramsey number $R(G,H)$ is defined as the minimum number of vertices $n$ such that every $\{red,blue\}$-edge-coloring of $K_n$ contains either a red copy of $G$ or a blue copy of $H$. If $G\cong H$, then we write $R\left(G\right)$ for short. For any positive integer $k$, the Gallai–Ramsey number ${gr}_k(G:H)$ is the minimum number of vertices $n$ such that any exact $k$-edge coloring of $K_n$ contains either a rainbow copy of $G$ or a monochromatic copy of $H$. In this paper, we give exact values or upper and lower bounds for Ramsey numbers $R\left(C{b}_n\right)$, $R\left(S_m\right)$, $R(S_m,C{b}_n)$ and Gallai–Ramsey numbers ${gr}_k(K_{1,3}:C{b}_n)$, ${gr}_k(K_{1,3}:S_m)$, where $C{b}_n$ and $S_m$ are the comb and sun graphs, respectively.