Entire colouring of plane graphs

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Abstract

It was conjectured by Kronk and Mitchem in 1973 that simple plane graphs of maximum degree Δ are entirely (Δ+4)-colourable, i.e., the vertices, edges, and faces of a simple plane graph may be simultaneously coloured with Δ+4 colours in such a way that adjacent or incident elements are coloured by distinct colours. Before this paper, the conjecture has been confirmed for Δ3 and Δ6 (the proof for the Δ=6 case has a correctable error). In this paper, we settle the whole conjecture in the positive. We prove that if G is a plane graph with maximum degree 4 (parallel edges allowed), then G is entirely 8-colourable. If G is a plane graph with maximum degree 5 (parallel edges allowed), then G is entirely 9-colourable.

Keywords

Entire colouring
Plane graph
Vertex–edge–face colouring

Cited by (0)

1

Supported partially by NSFC (No. 11071223), ZJNSF (No. Z6090150), and ZJIP (No. T200905).

2

Supported by ZJNSF (No. Z61107086).