Entire colouring of plane graphs

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 Δ ⩽ 3 and Δ ⩾ 6 (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.