Gallai-Ramsey number of odd cycles with chords

TL;DR

This paper determines the Gallai-Ramsey numbers for odd cycles with chords, providing a unified formula for all odd cycles with at least five vertices in Gallai colorings.

Contribution

It establishes an exact formula for the Gallai-Ramsey number of odd cycles with chords, extending previous results to a broader class of graphs.

Findings

Gallai-Ramsey number for $ heta_{2n+1}$ is $n imes 2^k + 1$

Gallai-Ramsey number for odd cycles $C_{2n+1}$ is $n imes 2^k + 1$

Unified proof for all odd cycles with at least five vertices

Abstract

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 $k\geq 1$, 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 $m\ge4$ vertices and let $\Theta_m$ denote the family of graphs obtained from $C_m$ by adding an additional edge joining two non-consecutive vertices. We prove that $GR_k(\Theta_{2n+1})=n\cdot 2^k+1$ for all $k\geq 1$ and $n\geq 3$. This implies that $GR_k(C_{2n+1})=n\cdot 2^k+1$ all $k\geq 1$ and $n\geq 3$. Our result yields a unified proof for the Gallai-Ramsey number of all odd cycles on at least five vertices.