Facial entire colouring of plane graphs

Let G=(V,E,F)G=(V,E,F) be a connected, loopless, and bridgeless plane graph, with vertex set VV, edge set EE, and face set FF. For X∈{V,E,F,V∪E,V∪F,E∪F,V∪E∪F}X∈{V,E,F,V∪E,V∪F,E∪F,V∪E∪F}, two elements xx and yy of XX are facially adjacent in GG if they are incident, or they are adjacent vertices, or adjacent faces, or facially adjacent edges (i.e. edges that are consecutive on the boundary walk of a face of GG). A kk-colouring is facial with respect to XX if there is a kk-colouring of elements of XX such that facially adjacent elements of XX receive different colours. We prove that: (i) Every plane graph G=(V,E,F)G=(V,E,F) has a facial 8-colouring with respect to X=V∪E∪FX=V∪E∪F (i.e. a facial entire 8-colouring). Moreover, there is plane graph requiring at least 7 colours in any such colouring. (ii) Every plane graph G=(V,E,F)G=(V,E,F) has a facial 6-colouring with respect to X=E∪FX=E∪F, in other words, a facial edge–face 6-colouring.