  For a graph $H$ and an integer $k\ge1$, the $k$-color Ramsey number $R_k(H)$
is the least integer $N$ such that every $k$-coloring of the edges of the
complete graph $K_N$ contains a monochromatic copy of $H$. Let $C_m$ denote the
cycle on $m\ge4$ vertices and let $\Theta_m$ denote the family of graphs
obtained from $C_m$ by adding an additional edge joining two non-consecutive
vertices. Unlike Ramsey number of odd cycles, little is known about the general
behavior of $R_k(C_{2n})$ except that $R_k(C_{2n})\ge (n-1)k+n+k-1$ for all
$k\ge2$ and $n\ge2$. In this paper, we study Ramsey number of even cycles with
chords under Gallai colorings, where a Gallai coloring is a coloring of the
edges of a complete graph without rainbow triangles. For an integer $k\geq 1$,
the Gallai-Ramsey number $GR_k(H)$ of a graph $H$ is the least positive integer
$N$ such that every Gallai $k$-coloring of the complete graph $K_N$ contains a
monochromatic copy of $H$. We prove that $GR_k(\Theta_{2n})=(n-1)k+n+1$ for all
$k\geq 2$ and $n\geq 3$. This implies that $GR_k(C_{2n})=(n-1)k+n+1$ all $k\geq
2$ and $n\geq 3$. Our result yields a unified proof for the Gallai-Ramsey
number of all even cycles on at least four vertices.
