Minimum span of no-hole (r+1)-distant colorings

Given a nonnegative integer r, a no-hole (r + 1)-distant coloring, called N -r-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 o f adjacent vertices is greater than r, and the set of the colors used must be consecutive. Given r and G, the minimum N-r-span of G, nsp(r)(G), is the minimum difference of the largest and the smallest colors used in an N-r-c oloring of G if there exists one; otherwise, define nsp,(G) = infinity. The values of nsp(1)(G) (r = 1) for bipartite graphs are given by Roberts [Mat h. Comput. Modelling, 17 (1993), pp. 139-144]. Given r greater than or equa l to 2, we determine the values of nsp(r)(G) for all bipartite graph with a t least r - 2 isolated vertices. This leads to complete solutions of nsp(2) (G) for bipartite graphs.