Bandwidth versus Bandsize

The bandwidth (bandsize) of a graph G is the minimum, over all bijections u: V(G) → {1,2,…,|V(G)|}, of the greatest difference (respectively the number of distinct differences) |u(v)—u(w)| for vw ɛ E(G). We show that a graph on n vertices with bandsize k has bandwidth between k and cn 1-1/n , and that this is best possible. In the process we obtain best possible asymptotic bounds on the bandwidth of circulant graphs. The bandwidth and bandsize of random graphs are also compared, the former turning out to be n — C 1 log n and the latter at least n —c 2 (logn) 2 .