Facial parity edge colouring

Abstract

A facial parity edge colouring of a connected bridgeless plane graph is an edge colouring in which no two face-adjacent edges (consecutive edges of a facial walk of some face) receive the same colour, in addition, for each face α and each colour c, either no edge or an odd number of edges incident with α is coloured with c. From Vizing's theorem it follows that every 3-connected plane graph has a such colouring with at most ∆ * + 1 colours, where ∆ * is the size of the largest face. In this paper we prove that any connected bridgeless plane graph has a facial parity edge colouring with at most 92 colours.

Improved bounds for some facially constrained colorings

“…We denote with χ f p (G) the minimum number k, for which there exists a facial-parity edge-coloring of G with k colors. Facialparity edge-coloring was first studied by Czap, Jendroľ and Kardoš in [8], where they proved that 92 colors suffice to color every 2-edge-connected plane graph. This bound was later improved by Czap et al [9] to 20 colors.…”

On the strong parity chromatic number

“…Assume that G contains at least two non-triangle faces which influence each other. Let the face f i ∈ B 4 be incident with the vertices ¿From Lemma 13 it follows that H 4 is a planar graph, hence, we can assign to each vertex of H one pair of colours from {(5, 6), (7,8), (9,10), (11,12)} in such a way that two adjacent vertices receive distinct pairs. It means that we can assign a pair of colours to each face of G of size at least four in such a way that two faces which influence each other receive distinct pairs.…”

Improved bound on facial parity edge coloring

“…In [2], Czap, Jendrol', and Kardoš defined the FPE-coloring of plane graphs and proved that 92 colors suffice to color any 2-edge connected plane graph. Recently, Czap et al [3] improved the upper bound.…”