Gallai-Ramsey number of even cycles with chords

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