Facial edge–face coloring of [formula omitted]-minor-free graphs

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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 χ̄ef(G), is defined to be the least integer k such that G is facially edge–face k-colorable.In 2016, Fabrici, Jendrol’ and Vrbjarová 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 K4-minor-free graphs. More precisely, we prove that every bridgeless K4-minor-free graph is facially edge–face 5-colorable. Moreover, the upper bound 5 is best possible.