Triangle-free graphs with diameter 2

TL;DR

This paper investigates the existence and properties of triangle-free graphs with diameter 2, exploring conditions under which infinitely many such graphs exist, and presents partial theoretical and computational results.

Contribution

It provides new partial results and constructions related to the existence of infinite families of triangle-free diameter-2 graphs with various forbidden subgraphs.

Findings

Finiteness results for certain girth and subgraph conditions.

Computational evidence supporting the existence of specific graph families.

Potential constructions using Cayley graphs and probabilistic methods.

Abstract

There are finitely many graphs with diameter $2$ and girth 5. What if the girth 5 assumption is relaxed? Apart from stars, are there finitely many triangle-free graphs with diameter $2$ and no $K_{2,3}$ subgraph? This question is related to the existence of triangle-free strongly regular graphs, but allowing for a range of co-degrees gives the question a more extremal flavour. More generally, for fixed $s$ and $t$, are there infinitely many twin-free triangle-free $K_{s,t}$-free graphs with diameter 2? This paper presents partial results regarding these questions, including computational results, potential Cayley-graph and probabilistic constructions.