Gallai-Ramsey numbers for graphs with five vertices and eight edges

Abstract

A Gallai $k$-coloring is a $k$-edge coloring of a complete graph in which there are no rainbow triangles. For given graphs $G_1, G_2, G_3$ and nonnegative integers $r, s, t$ with that $k=r+s+t$, the $k$-colored Gallai-Ramsey number $gr_{k}(K_{3}: r\cdot G_1,~ s\cdot G_2, ~t\cdot G_3)$ is the minimum integer $n$ such that every Gallai $k$-colored $K_{n}$ contains a monochromatic copy of $G_1$ colored by one of the first $r$ colors or a monochromatic copy of $G_2$ colored by one of the middle $s$ colors or a monochromatic copy of $G_3$ colored by one of the last $t$ colors. In this paper, we determine the value of Gallai-Ramsey number in the case that $G_1=B_{3}^{+}$, $G_2=S_{3}^+$ and $G_3=K_3$. Then the Gallai-Ramsey number $gr_{k}(K_{3}: B_{3}^{+})$ is obtained. Thus the Gllai-Ramsey numbers for graphs with five vertices and eight edges are solved completely. Furthermore, the the Gallai-Ramsey numbers $gr_{k}(K_{3}: r\cdot B_3^+,~ (k-r)\cdot S_3^+)$, $gr_{k}(K_{3}: r\cdot B_3^+,~ (k-r)\cdot K_3)$ and $gr_{k}(K_{3}: s\cdot S_3^+,~ (k-s)\cdot K_3)$ are obtained, respecticely.