Approximations for λ-colorings of graphs

A lambda-coloring of a graph G is an assignment of colors from the integer set {0,...,lambda} to the vertices of the graph G such that vertices at distance of at most two get different colors and adjacent vertices get colors which are at least two apart. The problem of finding lambda-colorings with optimal or near-optimal lambda arises in the context of radio frequency assignment. We show that the problem of finding the minimum lambda for planar graphs, bipartite graphs, chordal graphs and split graphs is NP-complete. We also give approximation algorithms for lambda-coloring and compute upper bounds on the best possible lambda for outerplanar graphs, graphs of treewidth k, permutation and split graphs. Except in the case of split graphs, all the above bounds for lambda are linear in Delta, the maximum degree of the graph. For split graphs, we give a bound of 1/2Delta(1.5) + 2Delta and we show that there are split graphs G with lambda(G)=Omega (Delta(1.5)). Similar results are also given for variations of the lambda-coloring problem.