  A Gallai coloring of a complete graph is an edge-coloring such that no
triangle has all its edges colored differently. A Gallai $k$-coloring is a
Gallai coloring that uses $k$ colors. Given a graph $H$ and an integer $k\geq
1$, the Gallai-Ramsey number $GR_k(H)$ of $H$ is the least positive integer $N$
such that every Gallai $k$-coloring of the complete graph $K_N$ contains a
monochromatic copy of $H$. Let $W_{2n} $ denote an even wheel on $2n+1\ge5$
vertices. In this note, we study Gallai-Ramsey number of $W_{2n}$ and
completely determine the exact value of $GR_k(W_4)$ for all $k\ge2$.
