Gallai-Ramsey numbers for monochromatic $K_4^{+}$ or $K_{3}$

TL;DR

This paper determines the Gallai-Ramsey number for monochromatic copies of the graph $K_4^+$ or $K_3$ in Gallai $k$-colored complete graphs, extending understanding of colorings avoiding rainbow triangles.

Contribution

The paper explicitly computes the Gallai-Ramsey number for $K_4^+$ versus $K_3$, a previously unresolved case in Gallai coloring theory.

Findings

Exact value of $gr_k(K_3: K_4^+)$ obtained.

Provides new insights into Gallai colorings avoiding rainbow triangles.

Extends known results in Gallai-Ramsey theory.

Abstract

A Gallai $k$-coloring is a $k$-edge coloring of a complete graph in which there are no rainbow triangles. For two given graphs $H, G$ and two positive integers $k,s$ with that $s\leq k$, the $k$-colored Gallai-Ramsey number $gr_{k}(K_{3}: s\cdot H,~ (k-s)\cdot G)$ is the minimum integer $n$ such that every Gallai $k$-colored $K_{n}$ contains a monochromatic copy of $H$ colored by one of the first $s$ colors or a monochromatic copy of $G$ colored by one of the remaining $k-s$ colors. In this paper, we determine the value of Gallai-Ramsey number in the case that $H=K_{4}^{+}$ and $G=K_{3}$. Thus the Gallai-Ramsey number $gr_{k}(K_{3}: K_{4}^{+})$ is obtained.