The homomorphism threshold of $\{C_3, C_5\}$-free graphs

TL;DR

This paper characterizes the structure of certain triangle and pentagon-free graphs with high minimum degree, showing they are homomorphic to a specific cycle-based graph, and determines their homomorphism threshold as 1/5.

Contribution

It precisely identifies the structure of 3, C5-free graphs with high minimum degree and establishes their homomorphism threshold as 1/5.

Findings

Graphs with minimum degree > n/5 are homomorphic to a specific cycle-based graph.

The homomorphism threshold for 3, C5-free graphs is 1/5.

Answers to open questions by Messuti, Schacht, Oberkampf, and Schacht.

Abstract

We determine the structure of $\{C_3, C_5\}$-free graphs with $n$ vertices and minimum degree larger than $n/5$: such graphs are homomorphic to the graph obtained from a $(5k - 3)$-cycle by adding all chords of length $1$ mod $5$, for some $k$. This answers a question of Messuti and Schacht. We deduce that the homomorphism threshold of $\{C_3, C_5\}$-free graphs is $1/5$, thus answering a question of Oberkampf and Schacht.