# 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 facially-proper vertex-coloring in which every face is incident with zero or an odd number of vertices of each color. Czap and Jendroľ (in Facially-constrained colorings of plane graphs: A survey, 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 (P_k, P_ℓ)-WORM coloring of a plane graph G is a coloring of the vertices such that G contains no rainbow facial k-path and no monochromatic facial ℓ-path. Czap, Jendroľ and Valiska (in WORM colorings of planar graphs, Discuss. Math. Graph Theory 37 (2017), 353–368), proved that for any integer n≥ 12 there exists a connected plane graph on n vertices, with maximum degree at least 6, having no facial (P_3,P_3)-WORM coloring. They also asked if there exists a graph with maximum degree 4 having the same property. We prove that for any integer n≥ 18, there exists a connected plane graph, with maximum degree 4, with no facial (P_3,P_3)-WORM coloring.

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