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 that the diameter of SG(2k + r, k) is equal to 2 if r ≥ 2k - 2; 3 if k≥ - 2 ≤ r ≤ 2k - 3; k if r = 1; and for 2 ≤ r ≤ k - 3, we obtain that the diameter of SG(2k + r, k) is at most equal to k - r + 1.