Asymmetric results about graph homomorphisms

TL;DR

The paper investigates asymmetric questions about graph homomorphisms, focusing on when a large F-free graph G can be mapped to a small F-free Γ under weaker or different assumptions on G and Γ. It introduces asymmetric homomorphism thresholds δ_hom(𝓕1;𝓕2) and establishes zero-threshold results in several regimes, notably showing that a 𝓒_{2t+5}-free graph with minimum degree δ|G| admits a homomorphism to a constant-size 𝓒_{2t+1}-free graph. It then studies asymmetric approximate homomorphisms through M_{F,H}(ε), revealing a diverse spectrum: super-exponential, exponential, and linear behavior depending on ℓ in C_ℓ, with a subdivision-based polynomial bound when H→F^{◦◦}. The work combines entropy arguments, the Frieze–Kannan weak regularity lemma, generalized Mycielskians, and high-girth hypergraph constructions to derive upper and lower bounds, and it highlights a double phase transition phenomenon in the bounds. These results enhance understanding of how triangle-freeness and related forbidden configurations can be explained by small, structured graphs, with implications for algorithmic construction and removal-lemma-type tools in asymmetric settings.

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 $Γ$ 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 $Γ$ need not match. For example, we prove that if $G$ is a graph with odd girth at least $9$ and minimum degree at least $δ|G|$, then $G$ is homomorphic to a triangle-free graph whose size depends only on $δ$. 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.

Figures (3)

• Figure 1: Illustration of \ref{['con:HG']}
• Figure 2: An example of a $K_3$-forest
• Figure 3: The $2$-fold Mycielskian $M_2(C_7)$. Note that this graph is $\mathscr{C}_5$-free, as implied by \ref{['lem:t-fold odd girth']}.

Theorems & Definitions (48)

• Theorem 1.1: Ł uczak MR2260851
• Theorem 1.2: Approximate homomorphism theorem
• Theorem 1.3
• Conjecture 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7: Graph removal lemma MR519318MR1251840MR1404036
• Definition 1.8
• Theorem 1.9
• Proposition 1.10