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 \((P_{k}, P_{\ell})\)-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 \(\ell\)-path. Czap, Jendroľ and Valiska (in WORM colorings of planar graphs, Discuss. Math. Graph Theory 37 (2017), 353--368), proved that for any integer \(n\ge 12\) 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 \(n\ge 18\), there exists a connected plane graph, with maximum degree \(4\), with no facial \((P_{3},P_{3})\)-WORM coloring.