Efficient Algorithms for Acyclic Colorings of Graphs

An acyclic k-coloring of a graph G is a coloring of the vertices of G with at most k colors such that each color class induces an acyclic subgraph. The vertex arboricity a(G) of G is the minimum number k for which G has an acyclic k-coloring. Although the problem of computing a(G) is NP-hard, ρ(G)=1+⌊(maxδ(G′))/2⌋ is known to be a good upper bound on a(G), where the maximum is taken over all induced subgraphs G′ of G and δ(G′) is the minimum degree of G′. In this paper, we present the first linear-time algorithm for acyclic ρ(G)-colorings. We also give a sufficient condition under which an NC algorithm exists for acyclic ρ(G)-colorings. Using this condition, we obtain the first NC algorithm for acyclic ρ(G)-colorings of graphs without a K3,3 (or K5) minor.