Asymmetric results about graph homomorphisms

TL;DR

This paper explores asymmetric graph homomorphism results, establishing new bounds and conditions under which graphs with certain properties can be homomorphically mapped to smaller, triangle-free graphs, with implications for extremal graph theory.

Contribution

It introduces asymmetric versions of homomorphism theorems, providing improved bounds and conditions involving odd girth, VC dimension, and domination number, with novel proof techniques.

Findings

Graphs with high minimum degree and odd girth at least 9 are homomorphic to smaller triangle-free graphs.

Bounds on homomorphism thresholds exhibit a double phase transition from super-exponential to linear.

Weaker odd girth assumptions are sufficient when graphs have bounded VC dimension or domination number.

Abstract

Many important results in extremal graph theory can be roughly summarised as "if a triangle-free graph $G$ has certain properties, then it has a homomorphism to a triangle-free graph $\Gamma$ of bounded size". For example, bounds on homomorphism thresholds give such a statement if $G$ has sufficiently high minimum degree, and the approximate homomorphism theorem gives such a statement for all $G$, if one weakens the notion of homomorphism appropriately. In this paper, we study asymmetric versions of these results, where the assumptions on $G$ and $\Gamma$ need not match. For example, we prove that if $G$ is a graph with odd girth at least $9$ and minimum degree at least $\delta |G|$, then $G$ is homomorphic to a triangle-free graph whose size depends only on $\delta$. Moreover, the odd girth assumption can be weakened to odd girth at least $7$ if $G$ has bounded VC dimension or bounded domination number. This gives a new and improved proof of a result of Huang et al. We also prove that in the asymmetric approximate homomorphism theorem, the bounds exhibit a rather surprising ``double phase transition'': the bounds are super-exponential if $G$ is only assumed to be triangle-free, they become exponential if $G$ is assumed to have odd girth $7$ or $9$, and become linear if $G$ has odd girth at least $11$. Our proofs use a wide variety of techniques, including entropy arguments, the Frieze--Kannan weak regularity lemma, properties of the generalised Mycielskian construction, and recent work on abundance and the asymmetric removal lemma.