It is known that the acyclic chromatic number of a subcubic graph is at most four, and its acyclic edge chromatic number is at most five. We present algorithms that prove these two facts. Let n be the number of vertices of a graph, Our first algorithm takes O(n) time and uses four colors to properly color the vertices of any subcubic graph so that there is no 2-colored cycle. Our second algorithm takes O(n) time and uses five colors to properly color the edges of any subcubic graph so that there is no 2-colored cycle. Both are the first linear-time algorithms for the problems they solve.

We present a linear time algorithm to properly color the edges of any graph of maximum degree 3 using 4 colors. Our algorithm uses a greedy approach and utilizes a new structure theorem for such graphs.

We present a new proof of a theorem of Erdős, Rubin, and Taylor, which states that the list chromatic number (or choice number) of a connected, simple graph that is neither complete nor an odd cycle does not exceed its maximum degree Δ. Our proof yields the first-known linear-time algorithm to Δ-list-color graphs satisfying the hypothesis of the theorem. Without change, our algorithm can also be used to Δ-color such graphs. It has the same running time as, but seems to be much simpler than, the current known algorithm, due to Lovász, for Δ-coloring such graphs. We also give a specialized version of our algorithm that works on subcubic graphs (ones with maximum degree three) by exploiting a simple decomposition principle for them.

We present efficient algorithms for three coloring problems on subcubic graphs (ones with maximum degree 3). These algorithms are based on a simple decomposition principle for subcubic graphs. The first algorithm is for 4-edge coloring, or more generally, 4-list-edge coloring. Our algorithm runs in linear time, and appears to be simpler than previous ones. As evidence we give the first randomized EREW PRAM algorithm that uses O(n/log n) processors and runs in O(log n) time with high probability, where n is the number of vertices of the input graph. The second algorithm is the first linear-time algorithm to 5-total-color subcubic graphs. The third algorithm generalizes this to the first linear-time algorithm to 5-list-total-color subcubic graphs.