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Given a graph $H$ and an integer $k\ge1$, the Gallai-Ramsey number $GR_k(H)$is defined to be the minimum integer $n$ such that every $k$-edge coloring ofthe complete graph $K_n$ contains either a rainbow (all different colored)triangle or a monochromatic copy of $H$. In this paper, we study Gallai-Ramseynumbers for graphs with chromatic number three such as $\widehat{K}_m$ for$m\ge2$, where $\widehat{K}_m$ is a kipas with $m+1$ vertices obtained from thejoin of $K_1$ and $P_m$, and a class of graphs with five vertices, denoted by$\mathscr{H}$. We first study the general lower bound of such graphs andpropose a conjecture for the exact value of $GR_k(\widehat{K}_m)$. Then we givea unified proof to determine the Gallai-Ramsey numbers for many graphs in$\mathscr{H}$ and obtain the exact value of $GR_k(\widehat{K}_4)$ for $k\ge1$.Our outcomes not only indicate that the conjecture on $GR_k(\widehat{K}_m)$ istrue for $m\le4$, but also imply several results on $GR_k(H)$ for some $H\in\mathscr{H}$ which are proved individually in different papers.

Gallai-Ramsey numbers for graphs with chromatic number three