Every plane graph of maximum degree 8 has an edge-face 9-colouring

TL;DR

This paper proves that every plane graph with maximum degree 8 can be edge-face coloured using 9 colours, extending previous results for higher degrees and filling a gap in graph colouring theory.

Contribution

The authors extend Borodin's edge-face colouring bound to include plane graphs with maximum degree 8, a case not previously established.

Findings

Edge-face colouring with 9 colours is possible for maximum degree 8 graphs.

The result completes the extension of Borodin's bound down to degree 8.

The proof advances understanding of colourings in planar graphs.

Abstract

An edge-face colouring of a plane graph with edge set $E$ and face set $F$ is a colouring of the elements of $E \cup F$ such that adjacent or incident elements receive different colours. Borodin proved that every plane graph of maximum degree $\Delta\ge10$ can be edge-face coloured with $\Delta+1$ colours. Borodin's bound was recently extended to the case where $\Delta=9$. In this paper, we extend it to the case $\Delta=8$.