Barnette图的距离2染色
Distance-Two Colorings of Barnette Graphs
摘要：
巴内特确定了两个有趣的类立方多面体图，他证明存在一个汉密尔顿周期。古迪证明了两个类的交集的猜想。我们从距离二着色的角度来研究这些类。图$G$的距离2 $r$ -染色是指将$r$颜色分配给$G$的顶点，使得距离至多为2的任意两个顶点具有不同的颜色。请注意，一个三次图至少需要四种颜色。三次平面图的距离为2的四色问题是NP-完全的。我们声称这个问题仍然是NP-完全的三连通二部三次平面图，我们称之为第一类巴内特图，因为他们是第一类巴内特确定。相比之下，我们声称这个问题是多项式的三次平面图的面大小$3, 4, 5,$或$6$，我们称之为类型2巴内特图，因为他们的关系巴内特的第二个猜想。我们称古迪图为二型巴内特图，其所有面的大小都是$4$或$6$。我们充分描述了所有的Goodey图，承认一个距离2四着色，并根据他们的脸的大小来描述其余的类型2巴内特图，承认一个距离2四着色。对于四次平面图，第二类Barnette图的类似物是面大小为$3$或$4$的图。对于这个类，距离为2的四色问题也是多项式的;事实上，我们可以再次完全描述所有可着色的实例--正好有两个这样的图。
英文摘要：
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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