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A stability theorem for multi-partite graphs

Part of: Graph theory

Published online by Cambridge University Press:  11 August 2025

Wanfang Chen ,
Changhong Lu  and
Long-Tu Yuan
Wanfang Chen
Affiliation:
School of Mathematical Sciences Key Laboratory of MEA(Ministry of Education) Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, China
Changhong Lu
Affiliation:
School of Mathematical Sciences Key Laboratory of MEA(Ministry of Education) Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, China
Long-Tu Yuan*
Affiliation:
School of Mathematical Sciences Key Laboratory of MEA(Ministry of Education) Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, China
*
Corresponding author: Long-Tu Yuan; Email: ltyuan@math.ecnu.edu.cn
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Abstract

The Erdős–Simonovits stability theorem is one of the most widely used theorems in extremal graph theory. We obtain an Erdős–Simonovits type stability theorem in multi-partite graphs. Different from the Erdős–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 from the extremal graphs for $H$. As applications, we strengthen a theorem of Bollobás, Erdős, and Straus and solve a conjecture in a stronger form 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

StabilityTurán numberdisjoint cliquesmultipartite graph

MSC classification

Primary: 05C35: Extremal problems

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Type
Paper
Information
Combinatorics, Probability and Computing , First View , pp. 1 - 27
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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