Distance-Two Colorings of Barnette Graphs
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Barnette identified two interesting classes of cubic polyhedral graphs for which he conjectured the existence of a Hamiltonian cycle. Goodey proved the conjecture for the intersection of the two classes. We examine these classes from the point of view of distance-two colorings. A distance-two r-coloring of a graph G is an assignment of r colors to the vertices of G so that any two vertices at distance at most two have different colors. Note that a cubic graph needs at least four colors. The distance-two four-coloring problem for cubic planar graphs is known to be NP-complete. We claim the problem remains NP-complete for tri-connected bipartite cubic planar graphs, which we call type-one Barnette graphs, since they are the first class identified by Barnette. By contrast, we claim the problem is polynomial for cubic plane graphs with face sizes 3, 4, 5, or 6, which we call type-two Barnette graphs, because of their relation to Barnette's second conjecture. We call Goodey graphs those type-two Barnette graphs all of whose faces have size 4 or 6. We fully describe all Goodey graphs that admit a distance-two four-coloring, and characterize the remaining type-two Barnette graphs that admit a distance-two four-coloring according to their face size. For quartic plane graphs, the analogue of type-two Barnette graphs are graphs with face sizes 3 or 4. For this class, the distance-two four-coloring problem is also polynomial; in fact, we can again fully describe all colorable instances -- there are exactly two such graphs.
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