On Generalized Total Colourings of Planar Graphs

In this paper we study generalised total colourings of graphs where the colour classes formed by vertices and edges, respectively, induce forests, while incident edges/vertices receive distinct colours. In [M. Borowiecki and I. Broere, Hamiltonicity and Generalised Total Colourings of Planar Graphs, Discussiones Mathematicae Graph Theory 36 (2016) 243–257] it was conjectured that for planar graphs, four colours suffice for this type of colouring. We confirm this conjecture for two infinite families of planar graphs.