Off-Diagonal Commonality of Graphs via Entropy

TL;DR

This paper investigates the commonality of certain graphs in edge colorings of complete graphs, using an information-theoretic approach to identify new classes of graphs with this property, including some with uncommon components.

Contribution

It introduces a novel entropy-based method to analyze graph commonality and identifies new pairs of graphs that are (p,1-p)-common, expanding understanding of graph colorings.

Findings

Certain graphs from odd cycles and paths are shown to be common.

Existence of (p,1-p)-common pairs where one graph is uncommon.

Application of Schur convexity to strengthen common graph properties.

Abstract

A graph $H$ is common if the limit as $n\to\infty$ of the minimum density of monochromatic labelled copies of $H$ in an edge colouring of $K_n$ with red and blue is attained by a sequence of quasirandom colourings. We apply an information-theoretic approach to show that certain graphs obtained from odd cycles and paths via gluing operations are common. In fact, for every pair $(H_1,H_2)$ of such graphs, there exists $p\in(0,1)$ such that an appropriate linear combination of red copies of $H_1$ and blue copies of $H_2$ is minimized by a quasirandom colouring in which $p\binom{n}{2}$ edges are red; such a pair $(H_1,H_2)$ is said to be $(p,1-p)$-common. Our approach exploits a strengthening of the common graph property for odd cycles that was recently proved using Schur convexity. We also exhibit a $(p,1-p)$-common pair $(H_1,H_2)$ such that $H_2$ is uncommon.