Bandwidth of the cartesian product of two connected graphs

The bandwidth B(G) of a graph G is the minimum of the quantity max{|f(x)−f(y)|:xy∈E(G)} taken over all injective integer numberings f of G. The cartesian product of two graphs G and H, written as G×H, is the graph with vertex set V(G)×V(H) and with (u1,v1) adjacent to (u2,v2) if either u1 is adjacent to u2 in G and v1=v2 or u1=u2 and v1 is adjacent to v2 in H. In this paper we investigate the bandwidth of the cartesian product of two connected graphs. For a graph G, we denote the diameter of G and the connectivity of G by D(G) and κ(G), respectively. Let G and H be two connected graphs. Among other results, we show that if B(H)=κ(H) and |V(H)|⩾2B(H)D(G)−min{1,D(G)−1}, then B(G×H)=B(H)|V(G)|. Moreover, the order condition in this result is sharp.