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A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai $k$-coloring is a Gallai coloring that uses at most $k$ colors. For an integer $kgeq 1$, the Gallai-Ramsey number $GR_k(H)$ of a given 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 $H$. Let $C_m$ denote the cycle on $mge4$ vertices and let $Theta_m$ denote the family of graphs obtained from $C_m$ by adding an additional edge joining two non-consecutive vertices. We prove that $GR_k(Theta_{2n+1})=ncdot 2^k+1$ for all $kgeq 1$ and $ngeq 3$. This implies that $GR_k(C_{2n+1})=ncdot 2^k+1$ all $kgeq 1$ and $ngeq 3$. Our result yields a unified proof for the Gallai-Ramsey number of all odd cycles on at least five vertices.
For a graph $H$ and an integer $kge1$, 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 $mge4$ vert
Given a graph $G$ and a positive integer $k$, the emph{Gallai-Ramsey number} is defined to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) copy of $G$ or a monochromatic
The Ramsey number $r(H)$ of a graph $H$ is the minimum $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. The threshold Ramsey multiplicity $m(H)$ is then the minimum number of monochromatic
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 $kgeq 1$, the Gallai-Ramsey numbe
Given a positive integer $ r $, the $ r $-color size-Ramsey number of a graph $ H $, denoted by $ hat{R}(H, r) $, is the smallest integer $ m $ for which there exists a graph $ G $ with $ m $ edges such that, in any edge coloring of $ G $ with $ r $