Treewidth of the generalized Kneser graphs

TL;DR

This paper determines the exact treewidth of generalized Kneser graphs, extending previous results for Kneser graphs to cases where the intersection parameter t is at least 2, including the complement of Johnson graphs.

Contribution

It provides the exact treewidth of generalized Kneser graphs for t≥2 and large n, and specifically refines the treewidth of the complement of Johnson graphs for all n and k.

Findings

Exact treewidth of generalized Kneser graphs for t≥2 and large n.

Precise treewidth values for the complement of Johnson graphs.

Extension of known results from Kneser graphs to generalized cases.

Abstract

Let $n$, $k$ and $t$ be integers with $1\leq t< k \leq n$. The \emph{generalized Kneser graph} $K(n,k,t)$ is a graph whose vertices are the $k$-subsets of a fixed $n$-set, where two $k$-subsets $A$ and $B$ are adjacent if $|A\cap B|<t$. The graph $K(n,k,1)$ is the well-known \emph{Kneser graph}. In 2014, Harvey and Wood determined the exact treewidth of the Kneser graphs for large $n$ with respect to $k$. In this paper, we give the exact treewidth of the generalized Kneser graphs for $t\geq2$ and large $n$ with respect to $k$ and $t$. In the special case when $t=k-1$, the graph $K(n,k,k-1)$ usually denoted by $\overline{J(n,k)}$ which is the complement of the Johnson graph $J(n,k)$. We give a more precise result for the exact value of the treewidth of $\overline{J(n,k)}$ for any $n$ and $k$.