Total colourings of planar graphs with large girth

Abstract

It is proved that if G is a planar graph with total (vertex-edge) chromatic number χ″, maximum degree Δ and girth g, then χ″ = Δ + 1 if Δ ≥ 5 and g ≥ 5, or Δ ≥ 4 and g ≥ 6, or Α ≥ 3 and g ≥ 10. These results hold also for graphs in the projective plane, torus and Klein bottle. © 1998 Academic Press Limited.