Triangle-free graphs with diameter 2

Alice Devillers; Nina Kamčev; Brendan McKay; Padraig Ó Catháin; Gordon Royle; Geertrui Van de Voorde; Ian Wanless; David R. Wood

arXiv · 2406.00246

Triangle-free graphs with diameter 2

Alice Devillers·Nina Kamčev·Brendan McKay·Padraig Ó Catháin·Gordon Royle·Geertrui Van de Voorde·Ian Wanless·David R. Wood

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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.

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Alice Devillers, Nina Kamčev, Brendan McKay, Padraig Ó Catháin, Gordon Royle, Geertrui Van de Voorde, Ian Wanless, David R. Wood. 2024-06-01. Triangle-free graphs with diameter 2. https://arxiv.org/abs/2406.00246

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