λ-coloring of graphs

A λ-coloring of a graph G is an assignment of colors from the set {0,.,,λ} to the vertices of a graph G such that vertices at distance at most two get different colors and adjacent vertices get colors which are at least two apart. The problem of finding A-colorings with small or optimal A arises in the context of radio frequency assignment. We show that the problems of finding the minimum A for planar graphs, bipartite graphs, chordal graphs and split graphs are NP-Complete. We then give approximation algorithms for A-coloring and compute upperbounds of the best possible A for outerplanar graphs, planar graphs, graphs of treewidth k, permutation and split graphs. With the exception of the split graphs, all the above bounds for A are linear in Δ, the maximum degree of the graph. For split graphs, we give a bound of A < Δ 1.5 +2Δ+2 and show that there are split graphs with A = Ω(Δ 1.5 ). Similar results are also given for variations of the A-coloring problem.