A note on triangle-free graphs

TL;DR

This paper characterizes certain triangle-free graphs with high minimum degree, showing they are either a 5-cycle or a complete bipartite graph under specific conditions.

Contribution

It provides a classification of triangle-free graphs with high minimum degree that lack perfect matchings, identifying their exact isomorphism types.

Findings

Graphs are either $C_5$ or $K_{(n-1)/2,(n+1)/2}$

Conditions on minimum degree and absence of perfect matching determine graph structure

Results apply to graphs with at least 3 vertices

Abstract

We show that if $G$ is a simple triangle-free graph with $n\geq 3$ vertices, without a perfect matching, and having a minimum degree at least $\frac{n-1}{2}$, then $G$ is isomorphic either to $C_5$ or to $K_{\frac{n-1}{2},\frac{n+1}{2}}$.