Induced paths in strongly regular graphs

TL;DR

This paper characterizes the presence of induced paths P4, P5, and their complements in strongly regular graphs, providing elementary proofs and exploring specific graph families like Johnson, Kneser, and Hamming graphs.

Contribution

It offers an elementary proof for the existence of P4 in strongly regular graphs and investigates conditions for P5 and its complement in various well-known graph families.

Findings

A strongly regular graph contains an induced P4 if and only if it is primitive.

Conditions for induced P5 and its complement are identified in specific graph families.

The study includes analysis of Johnson, Kneser, Hamming, Latin square, and Steiner triple system graphs.

Abstract

This paper studies induced paths in strongly regular graphs. We give an elementary proof that a strongly regular graph contains a path $P_4$ as an induced subgraph if and only if it is primitive, i.e. it is neither a complete multipartite graph nor its complement. Also, we investigate when a strongly regular graph has an induced subgraph isomorphic to $P_5$ or its complement, considering several well-known families including Johnson and Kneser graphs, Hamming graphs, Latin square graphs, and block-intersection graphs of Steiner triple systems.