Algorithms and Complexity
It is NP-complete to determine whether A(G) ≤ 3 (Kostochka 1978). Coleman & Cai (1986) showed that the decision variant of the problem is NP-complete even when G is a bipartite graph.
Gebremedhin et al. (2008) demonstrated that every proper vertex coloring of a chordal graph is also an acyclic coloring. Since chordal graphs can be optimally colored in O(n+m) time, the same is also true for acyclic coloring on that class of graphs.
A linear-time algorithm to acyclically color a graph of maximum degree ≤ 3 using 4 colors or fewer was given by Skulrattanakulchai (2004). Yadav & Satish (2008) describe a linear-time algorithm to acyclically color a graph of maximum degree ≤ 5 using 8 colors or fewer and also to color a graph of maximum degree ≤ 6 using 12 colors or fewer.
Read more about this topic: Acyclic Coloring
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