A clique version of the Erd\H{o}s-Gallai stability theorems

TL;DR

This paper generalizes the Erd ext{"o}s-Gallai theorems on cycles and paths by introducing a clique version, using a novel approach that combines Pósa's rotation lemma with Kopylov's technique, leading to new stability results.

Contribution

It presents a new clique-based generalization of the Erd ext{"o}s-Gallai stability theorems, offering alternative proofs and extending existing results.

Findings

Established a clique version of the Erd ext{"o}s-Gallai stability theorems

Provided alternative proofs for recent related results

Extended theorems to broader classes of graphs

Abstract

Combining P\'{o}sa's rotation lemma with a technique of Kopylov in a novel approach, we prove a generalization of the Erd\H{o}s-Gallai theorems on cycles and paths. This implies a clique version of the Erd\H{o}s-Gallai stability theorems and also provides alternative proofs for some recent results.