A stability theorem for multi-partite graphs

TL;DR

This paper extends the Erd ext{"o}s-Simonovits stability theorem to multi-partite graphs, showing near-extremal graphs have a structured form even if not close to extremal configurations, and solves a related conjecture.

Contribution

It introduces a new stability theorem for multi-partite graphs, broadening the applicability of extremal graph theory results.

Findings

Established a stability theorem for multi-partite graphs.

Proved that near-extremal graphs have a specific structure.

Solved a conjecture on maximum edges without disjoint cliques.

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