On the diameter of Schrijver graphs

Abstract

For k≥1 and n≥2k, the well known Kneser graph KG(n,k) has all k-element subsets of an n-element set as vertices; two such subsets are adjacent if they are disjoint. Schrijver constructed a vertex-critical subgraph SG(n,k) of KG(n,k) with the same chromatic number. In this paper, we compute the diameter of the graph SG(2k+r,k) with r≥1. We obtain an exact value of the diameter of SG(2k+r,k) when r∈{1,2} or when r≥k−3. For the remaining cases, when 3≤r≤k−4, we show that the diameter of SG(2k+r,k) belongs to the set {4,…,k−r−1}.