A note on class one graphs with maximum degree six

It is known that there are class two graphs with Delta = 6 which can be embedded in a surface Sigma with Euler characteristic chi(Sigma) <= 0. However, it is unknown whether there are class two graphs on the projective plane or on the plane with Delta = 6. In this paper, we prove that every graph with Delta = 6 is class one if it can be embedded in a surface with Euler characteristic at least -3 and is C-3-free, or C-4-free or if it can be embedded in a surface with Euler characteristic at least -1 and is C-5-free. This generalizes Zhou's results in [G. Zhou, A note on graphs of class I, Discrete Math. 263 (2003) 339-345] on planar graphs. (c) 2006 Elsevier B.V. All rights reserved.