Triangle-free graphs with diameter 2
Alice Devillers, Nina Kam\v{c}ev, Brendan McKay, Padraig \'O, Cath\'ain, Gordon Royle, Geertrui Van de Voorde, Ian Wanless and, David R. Wood

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 and girth 5. What if the girth 5 assumption is relaxed? Apart from stars, are there finitely many triangle-free graphs with diameter and no 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 and , are there infinitely many twin-free triangle-free -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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Taxonomy
TopicsAdvanced Graph Theory Research · graph theory and CDMA systems · Computational Geometry and Mesh Generation
