Approximations for λ-Coloring of Graphs

A λ-coloring of a graph G is an assignment of colors from the integer set {0, . . . , λ} to the vertices of the 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-coloring with small or optimal λ arises in the context of radio frequency assignment. We show that the problem 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 upper bounds of the best possible λ for outerplanar 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 ∆, the maximum degree of the graph. For split graphs, we give a bound of λ ≤ ∆ + 2∆ + 2 and show that there are split graphs with λ = Ω(∆). We also give a bound of λ = Ω(∆) for bipartite graphs. Similar results are also given for variations of the λ-coloring problem.