Sequences of radius k for complete bipartite graphs

A k-radius sequence for a graph G is a sequence of vertices of G (typically with repetitions) such that for every edge uv of G vertices u and v appear at least once within distance k in the sequence. The length of a shortest k-radius sequence for G is denoted by fk(G). We give an asymptotically tight estimation on fk(G) for complete bipartite graphs which matches a lower bound, valid for all bipartite graphs. We also show that determining fk(G) for an arbitrary graph G is NP-hard for every constant k>1.