Abstract Let G be a plane graph. Two edges are said to be facially adjacent if they are consecutive on the boundary walk of a face of G . We call G facially edge–face k -colorable if there is a mapping from E ( G ) ∪ F ( G ) to a k color set so that any two facially adjacent edges, adjacent faces, and incident edge and face receive distinct colors. The facial edge–face chromatic number of G , denoted by χ e f ( G ) , is defined to be the least integer k such that G is facially edge–face k -colorable. In 2016, Fabrici, Jendrol’ and Vrbjarova conjectured that every connected, loopless, bridgeless plane graph is facially edge–face 5-colorable. In this paper, we confirm this conjecture for the case of K 4 -minor-free graphs. More precisely, we prove that every bridgeless K 4 -minor-free graph is facially edge–face 5-colorable. Moreover, the upper bound 5 is best possible.

中文翻译：

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K4-minor-free 图的面部边缘-面部着色

摘要 令 G 为平面图。如果两条边在 G 的面的边界步行上是连续的，则称两条边是面相邻的。如果存在从 E ( G ) ∪ F ( G ) 到 ak 颜色集的映射，使得任何两个面部相邻的边、相邻的面以及入射边和面都接收不同的颜色，我们称 G 面边-面 k 可着色。G 的面部边缘 - 面部色数，由 χ ef ( G ) 表示，定义为最小整数 k，使得 G 是面部边缘 - 面部 k 可着色的。2016 年，Fabrici、Jendrol' 和 Vrbjarova 推测每个连接的、无环的、无桥的平面图都是面边-面 5 可着色的。在本文中，我们在 K 4 -minor-free 图的情况下证实了这一猜想。更准确地说，我们证明了每个无桥 K 4 -minor-free 图都是面部边缘 - 面部 5 可着色的。而且，