A stability theorem for multi-partite graphs
Combinatorics
2026-01-14 v1
Abstract
The Erd\H{o}s-Simonovits stability theorem is one of the most widely used theorems in extremal graph theory. We obtain an Erd\H{o}s-Simonovits type stability theorem in multi-partite graphs. Different from the Erd\H{o}s-Simonovits stability theorem, our stability theorem in multi-partite graphs says that if the number of edges of an -free graph is close to the extremal graphs for , then has a well-defined structure but may be far away to the extremal graphs for . As an application, we solve a conjecture posed by Han and Zhao concerning the maximum number of edges in multi-partite graphs which does not contain vertex-disjoint copies of a clique
Keywords
Cite
@article{arxiv.2208.13995,
title = {A stability theorem for multi-partite graphs},
author = {Wanfang Chen and Changhong Lu and Long-Tu Yuan},
journal= {arXiv preprint arXiv:2208.13995},
year = {2026}
}