Antibandwidth and cyclic antibandwidth of meshes and hypercubes

The antibandwidth problem consists of placing the vertices of a graph on a line in consecutive integer points in such a way that the minimum difference of adjacent vertices is maximized. The problem was originally introduced in [15] in connection with multiprocessor scheduling problems and can be also understood as a dual problem to the well known bandwidth problem, as a special radiocolouring problem or as a variant of obnoxious facility location problems. The antibandwidth problem is NP-hard, there are a few classes of graphs with polynomial time complexities. Exact results for nontrivial graphs are very rare. Miller and Pritikin [18] showed tight bounds for 2-dimensional meshes and hypercubes. We solve the antibandwidth problem precisely for two dimensional meshes, tori and estimate the antibandwidth value for hypercubes up to the third order term. The cyclic antibandwidth problem is to embed an n-vertex graph into the cycle Cn, such that the minimum distance (measured in the cycle) of adjacent vertices is maximised. This is a natural extension Preprint submitted to Elsevier Science 27 October 2005 of the antibandwidth problem or a dual problem to the cyclic bandwidth problem. We start investigating this invariant for typical graphs and prove basic facts and exact results for the same product graphs as for the antibandwidth.