Asymmetric Results About Graph Homomorphisms

ABSTRACT

Many important results in extremal graph theory can be roughly summarized as “if a triangle‐free graph has certain properties, then it has a homomorphism to a triangle‐free graph of bounded size.” For example, bounds on homomorphism thresholds give such a statement if has sufficiently high minimum degree, and the approximate homomorphism theorem gives such a statement for all if one weakens the notion of homomorphism appropriately. In this paper, we study asymmetric versions of these results, where the assumptions on and need not match. For example, we prove that if is a graph with odd girth at least 9 and minimum degree at least , then is homomorphic to a triangle‐free graph whose size depends only on . Moreover, the odd girth assumption can be weakened to odd girth at least 7 if has bounded VC dimension or bounded domination number. This gives a new and improved proof of a result of Huang, Liu, Rong, and Xu. 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 is only assumed to be triangle‐free, they become exponential if is assumed to have odd girth 7 or 9, and become linear if 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 generalized Mycielskian construction, and recent work on abundance and the asymmetric removal lemma.