Some locally Kneser graphs

TL;DR

This paper explores the structure of locally Kneser graphs, providing new examples for specific parameters and demonstrating the optimality of Hall's bound in characterizing such graphs.

Contribution

The authors identify new examples of locally Kneser graphs for n=3d and some sporadic cases, extending the understanding of their structure beyond Hall's original results.

Findings

New examples of locally Kneser graphs for n=3d

Confirmation that Hall's bound is tight

Extension of known classifications of locally Kneser graphs

Abstract

The Kneser graph $K(n,d)$ is the graph on the $d$-subsets of an $n$-set, adjacent when disjoint. Clearly, $K(n+d,d)$ is locally $K(n,d)$. Hall showed for $n \ge 3d+1$ that there are no further examples. Here we give other examples of locally $K(n,d)$ graphs for $n = 3d$, and some further sporadic examples. It follows that Hall's bound is best possible.