Quickly proving the Andr\'asfai-Erd\H{o}s-S\'os-Theorem

TL;DR

This paper presents a new, quicker proof of the Andre1sfalvi-Erd
51s-Sf3 theorem, which characterizes the structure of certain extremal graphs based on minimum degree and chromatic number.

Contribution

It introduces an alternative, more efficient proof technique for the Andre1sfalvi-Erd
51s-Sf3 theorem, enhancing understanding of extremal graph properties.

Findings

Provides a faster proof of the theorem

Clarifies the structure of extremal graphs

Strengthens theoretical understanding of graph colorability

Abstract

Given an integer $r\gs 2$, an important theorem first proved by B. Andr\'asfai, P. Erd\H{o}s, and V. T. S\'os states that any $K_{r+1}$--free graph on $n$ vertices whose minimum degree is greater than $(3r-4)n/(3r-1)$ is $r$--colourable, and determines the graphs that are extremal in this context. The purpose of this note is to give an alternative proof of this result using a different idea.