Bandwidth of graphs resulting from the edge clique covering problem

TL;DR

This paper investigates the bandwidth of a class of graphs derived from the edge clique covering problem, providing exact and asymptotic results for various parameter regimes and proposing conjectures for unresolved cases.

Contribution

It introduces a new graph class related to hypergraph bandwidth, determines their exact bandwidth for certain parameters, and provides asymptotic bounds in other regimes.

Findings

Exact bandwidth for $b \,\geq\, \frac{n+k-1}{2}$

Asymptotic bandwidth for $b=o(n)$ and linear $b$ with factor $eta$

Conjecture on the asymptotic value for the open case

Abstract

Let $n,k,b$ be integers with $1 \le k-1 \le b \le n$ and let $G_{n,k,b}$ be the graph whose vertices are the $k$-element subsets $X$ of $\{0,\dots,n\}$ with $\max(X)-\min(X) \le b$ and where two such vertices $X,Y$ are joined by an edge if $\max(X \cup Y) - \min(X \cup Y) \le b$. These graphs are generated by applying a transformation to maximal $k$-uniform hypergraphs of bandwidth $b$ that is used to reduce the (weak) edge clique covering problem to a vertex clique covering problem. The bandwidth of $G_{n,k,b}$ is thus the largest possible bandwidth of any transformed $k$-uniform hypergraph of bandwidth $b$. For $b\geq \frac{n+k-1}{2}$, the exact bandwidth of these graphs is determined. For $b<\frac{n+k-1}{2}$, the bandwidth is asymptotically determined in the case of $b=o(n)$ and in the case of $b$ growing linearly in $n$ with a factor $\beta \in (0,0.5]$, where for one case only bounds could be found. It is conjectured that the upper bound of this open case is the right asymptotic value.