Dichotomy for coloring of dart graphs

Abstract

We study a (k + 1)-coloring problem in a class of (k,s)-dart graphs, k,s ≥ 2, where each vertex of degree at least k + 2 belongs to a (k,i)-diamond, i ≤ s. We prove that dichotomy holds, that means the problem is either NP-complete (if k < s), or can be solved in linear time (if k ≥ s). In particular, in the latter case we generalize the classical Brooks Theorem, that means we prove that a (k, s)-dart graph, k ≥ max {2,s}, is (k + 1)-colorable unless it contains a component isomorphic to Kk + 2. © 2011 Springer-Verlag.