A note on class one graphs with maximum degree six

It is known that there are class two graphs with image which can be embedded in a surface image with Euler characteristic image. However, it is unknown whether there are class two graphs on the projective plane or on the plane with image. In this paper, we prove that every graph with image is class one if it can be embedded in a surface with Euler characteristic at least image and is image-free, or image-free or if it can be embedded in a surface with Euler characteristic at least image and is image-free. This generalizes Zhouʹs results in [G. Zhou, A note on graphs of class I, Discrete Math. 263 (2003) 339–345] on planar graphs.