Weighted variants of the Andr\'asfai-Erd\H{o}s-S\'os Theorem

TL;DR

This paper extends a classical extremal graph theory result to weighted graphs, exploring conditions under which weighted graphs avoid certain cliques and exhibit specific partition properties.

Contribution

It introduces weighted variants of the Andre1sfai-Erd
51s-Sf3 Theorem, expanding its applicability to weighted graphs and related Ramsey-Ture1n problems.

Findings

Established weighted conditions ensuring clique-free graphs are r-partite.

Connected weighted degree thresholds to clique avoidance.

Extended classical theorem to new weighted graph contexts.

Abstract

A well known result due to Andr\'asfai, Erd\H{o}s, and S\'os asserts that for $r\ge 2$ every $K_{r+1}$-free graph on $n$ vertices with $\delta(G)>\frac{3r-4}{3r-1}n$ is $r$-partite. We study related questions in the context of weighted graphs, which are motivated by recent work on the Ramsey-Tur\'an problem for cliques.