Homomorphism thresholds for odd cycles

TL;DR

This paper investigates the homomorphism thresholds in large graphs with forbidden odd cycles, focusing on the minimum degree conditions that guarantee a bounded, homomorphically related, $F$-free graph.

Contribution

It introduces the concept of homomorphism thresholds for odd cycles and advances understanding of their behavior in extremal graph theory.

Findings

Defined the homomorphism threshold for odd cycles.

Established bounds for the homomorphism threshold in odd cycle-free graphs.

Connected homomorphism thresholds to classical extremal graph parameters.

Abstract

The interplay of minimum degree conditions and structural properties of large graphs with forbidden subgraphs is a central topic in extremal graph theory. For a given graph $F$ we define the homomorphism threshold as the infimum over all $\alpha\in[0,1]$ such that every $n$-vertex $F$-free graph $G$ with minimum degree at least $\alpha n$ has a homomorphic image $H$ of bounded order (independent of $n$), which is $F$-free as well. Without the restriction of $H$ being $F$-free we recover the definition of the chromatic threshold, which was determined for every graph $F$ by Allen et al. [Adv. Math. 235 (2013), 261-295]. The homomorphism threshold is less understood and we address the problem for odd cycles.