Off-Diagonal Commonality of Graphs via Entropy

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2335-2360, September 2024.
Abstract. A graph [math] is common if the limit as [math] of the minimum density of monochromatic labeled copies of [math] in an edge coloring of [math] with red and blue is attained by a sequence of quasirandom colorings. 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 [math] of such graphs, there exists [math] such that an appropriate linear combination of red copies of [math] and blue copies of [math] is minimized by a quasirandom coloring in which [math] edges are red; such a pair [math] is said to be [math]-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 [math]-common pair [math] such that [math] is uncommon.