Edge number critical triangle free graphs with low independence numbers

TL;DR

This paper characterizes all triangle-free graphs with a specific edge-to-vertex relationship involving their independence number, answering a longstanding open question in graph theory.

Contribution

It provides a complete structural classification of certain triangle-free graphs based on a key edge-vertex-independence number relation, advancing understanding in extremal graph theory.

Findings

All such graphs are explicitly characterized.

The result confirms a conjecture posed by Radziszowski and Kreher.

The classification aids in understanding extremal properties of triangle-free graphs.

Abstract

The structure of all triangle free graphs G = (V,E) with |E| - 6|V| + 13\alpha(G) = 0 is determined, yielding an affirmative answer to a question of Stanis{\l}aw Radziszowski and Donald Kreher.