Strong Brandt-Thomass\'e Theorems

TL;DR

This paper extends a classic theorem on triangle-free graphs with high minimum degree, showing they have a specific blow-up structure under weaker conditions, advancing understanding in graph theory.

Contribution

It generalizes the Brandt-Thomassé theorem by weakening the degree condition while maintaining the structural conclusion for triangle-free graphs.

Findings

Maximal triangle-free graphs with certain neighbor sequence properties are blow-ups of Andre1sfai or Vega graphs.

The structural characterization holds under weaker assumptions than previously known.

Results will be applied to solve an open problem in Ramsey-Ture1n theory.

Abstract

Solving a long standing conjecture of Erd\H{o}s and Simonovits, Brandt and Thomass\'e proved that the chromatic number of each triangle-free graph $G$ such that $\delta(G)>|V(G)|/3$ is at most four. In fact, they showed the much stronger result that every maximal triangle-free graph $G$ satisfying this minimum degree condition is a blow-up of either an Andr\'asfai or a Vega graph. Here we establish the same structural conclusion on $G$ under the weaker assumption that for $m\in\{2, 3, 4\}$ every sequence of $3m$ vertices has a subsequence of length $m+1$ with a common neighbour. In forthcoming work this will be used to solve an old problem of Andr\'asfai in Ramsey-Tur\'an theory.