外平面图的弱完备染色

假设G=（V，E，F）是一个平面图。如果e₁和e₂是G中两条相邻边且在关联的面的边界上连续出现，那么称e₁和e₂面相邻。图G的一个弱完备k-染色是指存在一个从V ∪ E ∪ F到k色集合{1, …, K}的映射，使得任意两个相邻点，两个相邻面，两条面相邻的边，以及V ∪ E ∪ F中任意两个相关联的元素都染不同的颜色。若图G有一个弱完备k-染色，则称G是弱完备k-可染的。平面图G的弱完备色数是指G是弱完备k-可染的正整数k的最小值，记成χ_vef（G）。2016年，Fabrici等人猜想：每个无环且无割边的连通平面图是弱完备7-可染的。证明外平面图满足猜想，即外平面图是弱完备7-可染的。

Weakly entire coloring of outerplanar graphs

Let G=(V,E,F)be a plane graph.If e₁ and e₂ are consecutively adjacent with the same face,then we say that e₁ and e₂ are facially adjacent.A plane graph G is called weakly entire k-colorable if there is a mapping from V∪E∪F to{1,…,k}such that any facially adjacent edges,adjacent vertices,adjacent faces,and any two incident elements in V∪E∪F receive distinct colors.The weakly entire chromatic number,denoted X_vef(G),of G is defined to be the least integer k such that G is weakly entire k-colorable.In 2016,Fabrici et al.conjectured that every connected,loopless,bridgeless plane graph is weakly entire 7-colorable.In this paper,we prove that the conjecture is true for outerplanar graphs.Namely,outerplanar graphs are weakly entire 7-colorable.