Extremal problems and results related to Gallai-colorings

TL;DR

This paper investigates extremal problems in Gallai-colorings of complete graphs, providing bounds and exact values for edges and monochromatic triangles, and determining Gallai-Ramsey numbers for specific graphs.

Contribution

It establishes new bounds and exact values for extremal edge counts and monochromatic triangles in Gallai-colorings, and computes Gallai-Ramsey numbers for certain graphs.

Findings

Bounds for edges not in rainbow or monochromatic triangles.

Exact value of minimum monochromatic triangles for k=3.

Gallai-Ramsey number for K4+e.

Abstract

A Gallai-coloring (Gallai-$k$-coloring) is an edge-coloring (with colors from $\{1, 2, \ldots, k\}$) of a complete graph without rainbow triangles. Given a graph $H$ and a positive integer $k$, the $k$-colored Gallai-Ramsey number $GR_k(H)$ is the minimum integer $n$ such that every Gallai-$k$-coloring of the complete graph $K_n$ contains a monochromatic copy of $H$. In this paper, we consider two extremal problems related to Gallai-$k$-colorings. First, we determine upper and lower bounds for the maximum number of edges that are not contained in any rainbow triangle or monochromatic triangle in a $k$-edge-coloring of $K_n$. Second, for $n\geq GR_k(K_3)$, we determine upper and lower bounds for the minimum number of monochromatic triangles in a Gallai-$k$-coloring of $K_{n}$, yielding the exact value for $k=3$. Furthermore, we determine the Gallai-Ramsey number $GR_k(K_4+e)$ for the graph on five vertices consisting of a $K_4$ with a pendant edge.