Edge-colorings avoiding patterns in a triangle

TL;DR

This paper determines the maximum number of r-edge-colorings avoiding a triangle with exactly two colors for large graphs, showing bipartite Turán graphs are optimal for 2 to 26 colors.

Contribution

It establishes the extremal graphs for the maximum number of such colorings for 2 to 26 colors, extending understanding of pattern-avoiding colorings in large graphs.

Findings

Maximum is attained by bipartite Turán graph T_2(n) for 2 ≤ r ≤ 26.

T_2(n) is not extremal for r ≥ 27.

Results hold for sufficiently large n.

Abstract

For positive integers $n$ and $r$, we consider $n$-vertex graphs with the maximum number of $r$-edge-colorings with no copy of a triangle where exactly two colors appear. We prove that, if $2 \leq r \leq 26$ and $n$ is sufficiently large, the maximum is attained by the bipartite Tur\'{a}n graph $T_2(n)$ on $n$ vertices. This is best possible, as $T_2(n)$ is not extremal for $r \geq 27$ colors and $n \geq 3$.