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arXiv · 2208.13995

A stability theorem for multi-partite graphs

Wanfang Chen·Changhong Lu·Long-Tu Yuan
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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 $H$-free graph $G$ is close to the extremal graphs for $H$, then $G$ has a well-defined structure but may be far away to the extremal graphs for $H$. 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

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Wanfang Chen, Changhong Lu, Long-Tu Yuan. 2022-08-30. A stability theorem for multi-partite graphs. https://doi.org/10.1017/s0963548325100138

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