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。

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 χ_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.