Every Plane Graph of Maximum Degree 8 has an Edge-Face 9-Coloring

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 ∪ F such that adjacent or incident elements receive different colours. Borodin proved that every plane graph of maximum degree ∆ 10 can be edge-face coloured with ∆ + 1 colours. Borodin's bound was recently extended to the case where ∆ = 9. In this paper, we extend it to the case ∆ = 8.

Plane graphs of maximum degree Δ ≥ 7 are edge‐face (Δ + 1)‐colorable

“…By Theorem 1, every plane graph G with Δ=7 satisfies χ ef (G) ≤ 8. Combining with results in [1,5,11], we have established the following.…”

“…Borodin [1] proved that every plane graph G with Δ ≥ 10 is edge-face (Δ + 1)-colorable. The condition Δ ≥ 10 has recently been reduced to Δ = 9 [11] and further to Δ = 8 [5].…”