Asymptotic Determination of Edge-Bandwidth of Multidimensional Grids and Hamming Graphs

Abstract

The edge-bandwidth $B'(G)$ of a graph G is the bandwidth of the line graph of G. More specifically, for any bijection $f: E(G)\to \{1,2,\ldots, |E(G)|\}$, let $B'(f,G)=\max\{|f(e_1)- f(e_2)|: \mbox{$e1$ and $e2$ are incident edges of G}\}$, and let $B'(G)=\min_f B'(f,G)$. We determine asymptotically the edge-bandwidth of d-dimensional grids $P_n^d$ and of the Hamming graph $K_n^d$, the d-fold Cartesian product of $K_n$. Our results are as follows. (i) For fixed d and $n\to \infty$, $B'(P_n^d)=c(d)d n^{d-1}+O(n^{d-{3\over2}})$, where $c(d)$ is a constant depending on d, which we determine explicitly. (ii) For fixed even n and $d\to \infty$, $B'(K_n^d)=(1+o(1))\sqrt{d\over {2\pi}}\, n^d (n-1)$. Our results extend recent results by Balogh, Mubayi, and Pluhár [Theoret. Comput. Sci., 359 (2006), pp. 43–57], who determined $B'(P_n^2)$ asymptotically as a function of n and $B'(K_2^d)$ asymptotically as a function of d.