On the diameter of Schrijver graphs

TL;DR

This paper determines the diameter of Schrijver graphs SG(2k+r,k) for various parameters, providing exact values in some cases and bounds in others, advancing understanding of their structural properties.

Contribution

The paper computes the diameter of SG(2k+r,k) graphs, offering exact results for specific r values and bounds for others, extending previous knowledge of Schrijver graph properties.

Findings

Exact diameter for r  1, 2, or  k-3.

Diameter bounds for 3  r  k-4, between 4 and k-r-1.

Enhanced understanding of Schrijver graph structure.

Abstract

For $k \geq 1$ and $n \geq 2k$, the well known Kneser graph $\operatorname{KG}(n,k)$ 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 $\operatorname{SG}(n,k)$ of $\operatorname{KG}(n,k)$ with the same chromatic number. In this paper, we compute the diameter of the graph $\operatorname{SG}(2k+r,k)$ with $r \geq 1$. We obtain an exact value of the diameter of $\operatorname{SG}(2k+r,k)$ when $r \in \{1,2\}$ or when $r \geq k-3$. For the remained cases, when $3 \leq r \leq k-4$, we obtain that the diameter of $\operatorname{SG}(2k+r,k)$ belongs to the integer interval $[4..k-r-1]$.