Improved Bounds for Some Facially Constrained Colorings

A facial-parity edge-coloring of a 2-edge-connected plane graph is a facially-proper edge-coloring in which every face is incident with zero or an odd number of edges of each color. A facial-parity vertex-coloring of a 2-connected plane graph is a proper vertex-coloring in which every face is incident with zero or an odd number of vertices of each color. Czap and Jendroľ in [ , Discrete Math. 340 (2017) 2691–2703], conjectured that 10 colors suffice in both colorings. We present an infinite family of counterexamples to both conjectures. A facial ( )-WORM coloring of a plane graph is a vertex-coloring such that contains neither rainbow facial -path nor monochromatic facial -path. Czap, Jendroľ and Valiska in [ , Discuss. Math. Graph Theory 37 (2017) 353–368], proved that for any integer 12 there exists a connected plane graph on vertices, with maximum degree at least 6, having no facial ( , )-WORM coloring. They also asked whether there exists a graph with maximum degree 4 having the same property. We prove that for any integer 18, there exists a connected plane graph, with maximum degree 4, with no facial ( , )-WORM coloring.