Total Colourings of Planar Graphs with Large Girth

It is proved that ifGis a planar graph with total (vertex–edge) chromatic number χ″, maximum degree Δ and girthg, then χ″ = Δ+1 if Δ ≥ 5 andg ≥ 5, or Δ ≥ 4 andg ≥ 6, or Δ ≥ 3 andg ≥ 10. These results hold also for graphs in the projective plane, torus and Klein bottle.