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 λ-colorings with small or optimal λ arises in the context of radio frequency assignment. We show that the problems of finding the minimum λ for planar graphs, bipartite graphs, chordal graphs and split graphs are NP-Complete. We then give approximation algorithms for λ-coloring and compute upperbounds of the best possible λ 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 λ are linear in Δ, the maximum degree of the graph. For split graphs, we give a bound of λ ≤ Δ1.5+2Δ+2 and show that there are split graphs with λ = Ω(Δ1.5). Similar results are also given for variations of the λ-coloring problem.