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arXiv · 2607.28241

Exact Homomorphism Thresholds Beyond Cliques

Xinqi Huang·Mingyuan Rong·Chong Shangguan
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Abstract

The chromatic threshold, originating in a question of Erd\H{o}s and Simonovits, asks when a linear minimum-degree condition forces bounded chromatic number in H-free graphs. Motivated by a question of Thomassen, the homomorphism threshold asks for the stronger conclusion that every such graph admits a homomorphism to an H-free graph of bounded order. Since the work of Goddard and Lyle determined the clique case, exact homomorphism thresholds for individual non-complete forbidden graphs have remained unknown. In this paper, we extend the clique case to a larger family of forbidden graphs, determining the homomorphism threshold exactly for every graph in this family.

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Xinqi Huang, Mingyuan Rong, Chong Shangguan. 2026-07-30. Exact Homomorphism Thresholds Beyond Cliques. https://arxiv.org/abs/2607.28241

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