Minimum-degree thresholds beyond bounded colourability
Abstract
We survey recent developments on minimum-degree thresholds in extremal graph theory, with an emphasis on structural refinements of the chromatic-threshold problem. The chromatic threshold asks when a linear minimum-degree condition forces bounded chromatic number in an H-free graph; its complete classification shows a striking finite trichotomy. We discuss several stronger and related threshold notions in which bounded colourability is replaced by bounded H-free homomorphic images, bounded neighbourhood complexity via VC-dimension, exact blowup structure for maximal H-free graphs, and improved quantitative bounds in graph removal lemmas. In contrast with the rigidity of the chromatic threshold, these newer parameters exhibit richer behaviour, including separations from δχ(H), new exact values, accumulation points in fixed-chromatic spectra, positivity phenomena for 3-chromatic graphs, asymmetric variants, and failures of monotonicity.
Related
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BibTeX
@misc{huang2026minimumdegree,
title = {{Minimum-degree thresholds beyond bounded colourability}},
author = {Xinqi Huang and Hong Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39759},
eprint = {2610.39759},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39759},
}
RIS
TY - UNPB
TI - Minimum-degree thresholds beyond bounded colourability
AU - Xinqi Huang
AU - Hong Liu
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39759
UR - https://arxiv.org/abs/2610.39759
AB - We survey recent developments on minimum-degree thresholds in extremal graph theory, with an emphasis on structural refinements of the chromatic-threshold problem. The chromatic threshold asks when a linear minimum-degree condition forces bounded chromatic number in an H-free graph; its complete classification shows a striking finite trichotomy. We discuss several stronger and related threshold notions in which bounded colourability is replaced by bounded H-free homomorphic images, bounded neighbourhood complexity via VC-dimension, exact blowup structure for maximal H-free graphs, and improved quantitative bounds in graph removal lemmas. In contrast with the rigidity of the chromatic threshold, these newer parameters exhibit richer behaviour, including separations from δ_χ(H), new exact values, accumulation points in fixed-chromatic spectra, positivity phenomena for 3-chromatic graphs, asymmetric variants, and failures of monotonicity.
ER -