Extended commonality of paths and cycles via Schur convexity

Abstract

A graph $H$ is \emph{common} if the number of monochromatic copies of $H$ in a 2-edge-colouring of the complete graph $K_n$ is asymptotically minimised by the random colouring, or equivalently, $t_H(W)+t_H(1-W)\ge 2^{1-e(H)}$ holds for every graphon $W:[0,1{]}^2\to [0,1]$, where $t_H(.)$ denotes the homomorphism density of the graph $H$. Paths and cycles being common is one of the earliest cornerstones in extremal graph theory, due to Mulholland and Smith (1959), Goodman (1959), and Sidorenko (1989). We prove a graph homomorphism inequality that extends the commonality of paths and cycles. Namely, $t_H(W)+t_H(1-W)\ge t_{K_2}(W{)}^{e(H)}+t_{K_2}(1-W{)}^{e(H)}$ whenever $H$ is a path or a cycle and $W:[0,1{]}^2\to \mathbb{R}$ is a bounded symmetric measurable function. This answers a question of Sidorenko from 1989, who proved a slightly weaker result for even-length paths to prove the commonality of odd cycles. Furthermore, it also settles a recent conjecture of Behague, Morrison, and Noel in a strong form, who asked if the inequality holds for graphons $W$ and odd cycles $H$. Our proof uses Schur convexity of complete homogeneous symmetric functions, which may be of independent interest.

10 pages