Hamiltonicity and Generalised Total Colourings of Planar Graphs

The total generalised colourings considered in this paper are colourings of graphs such that the vertices and edges of the graph which receive the same colour induce subgraphs from two prescribed hereditary graph properties while incident elements receive different colours. The associated total chromatic number is the least number of colours with which this is possible. We study such colourings for sets of planar graphs and determine, in particular, upper bounds for these chromatic numbers for proper colourings of the vertices while the monochromatic edge sets are allowed to be forests. We also prove that if an even planar triangulation has a Hamilton cycle H for which there is no cycle among the edges inside H, then such a graph needs at most four colours for a total colouring as described above. The paper is concluded with some conjectures and open problems.