Discrete isoperimetric method for bandwidth, pathwidth and treewidth of hypercubes

Abstract

Let $\omega \left(G\right)$ denote the clique number of graph $G$. The treewidth $tw\left(G\right)$ of $G$ is the minimum of $\omega \left(H\right)-1$ taken over all chordal supergraph $H$ of $G$. The pathwidth $pw\left(G\right)$ and the bandwidth $bw\left(G\right)$ can be defined in a similar way when the chordal graph $H$ is replaced by an interval graph or a proper interval graph respectively. It follows that $tw\left(G\right)\le pw\left(G\right)\le bw\left(G\right)$. A $d$-dimensional hypercube $Q_d$ is the graph with vertex set of all $d$-tuples ${\alpha }_1{\alpha }_2\dots {\alpha }_d$ with ${\alpha }_i\in \{0,1\}$ where two vertices are adjacent if they differ in exactly one coordinate. In order to determine the bandwidth $\text{bw}\left(Q_d\right)$ of hypercubes, Harper (1966) proposed a powerful discrete isoperimetric method. Later, it was shown that $pw\left(Q_d\right)=bw\left(Q_d\right)$, but $tw\left(Q_d\right)$ is unknown so far. In this paper, we review the discrete isoperimetric method for $bw\left(Q_d\right)$, $pw\left(Q_d\right)$, and $tw\left(Q_d\right)$. In particular, we show that $tw\left(Q_d\right)=bw\left(Q_d\right)-1$ for $d\ge 3$ (it is trivial that $tw\left(Q_d\right)=bw\left(Q_d\right)$ for $d\le 2$).