Improved bounds for some facially constrained colorings

TL;DR

This paper investigates facially constrained colorings in plane graphs, providing counterexamples to existing conjectures and establishing new existence results for graphs with specific properties related to facial paths and colorings.

Contribution

The paper presents infinite families of counterexamples to conjectures on facial-parity colorings and proves the existence of certain plane graphs with maximum degree 4 lacking facial (P3, P3)-WORM colorings.

Findings

Counterexamples to conjectures on facial-parity colorings.

Existence of plane graphs with maximum degree 4 and no facial (P3, P3)-WORM coloring for n ≥ 18.

New bounds and constructions for facially constrained colorings in plane graphs.

Abstract

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 Jendro\v{l} (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, Jendro\v{l} 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.