Minimum Span of No-Hole (r+1)-Distant Colorings

Given a nonnegative integer r, a no-hole (r+1)-distant coloring, called Nr-coloring, of a graph G is a function that assigns a nonnegative integer (color) to each vertex such that the separation of the colors of any pair of adjacent vertices is greater than r, and the set of the colors used must be consecutive. Given r and G, the minimum Nr-span of G, nspr(G), is the minimum difference of the largest and the smallest colors used in an Nr-coloring of G if there exists one; otherwise, define nspr(G) = ∞. The values of nsp1(G) (r = 1) for bipartite graphs are given by Roberts [Math. Comput. Modelling, 17 (1993), pp. 139–144]. Given r ≥ 2, we determine the values of nspr(G) for all bipartite graph with at least r − 2 isolated vertices. This leads to complete solutions of nsp2(G) for bipartite graphs.