3-facial colouring of plane graphs

TL;DR

This paper proves that every plane graph can be coloured with 11 colours under 3-facial constraints, advancing understanding of facial colourings and related conjectures in graph theory.

Contribution

The authors establish that all plane graphs are 3-facially 11-colourable, providing new bounds close to longstanding conjectures in facial graph colouring.

Findings

Every plane graph is 3-facially 11-colourable

2-connected plane graphs with face-size ≤7 are cyclically 11-colourable

Bounds are near the (3l+1)-Conjecture and Cyclic Conjecture

Abstract

A plane graph is l-facially k-colourable if its vertices can be coloured with k colours such that any two distinct vertices on a facial segment of length at most l are coloured differently. We prove that every plane graph is 3-facially 11-colourable. As a consequence, we derive that every 2-connected plane graph with maximum face-size at most 7 is cyclically 11-colourable. These two bounds are for one off from those that are proposed by the (3l+1)-Conjecture and the Cyclic Conjecture.