Upper bounds on the maximum degree of class two graphs on surfaces

For each surface Sigma, we define Delta(Sigma)=max{Delta(G)vertical bar G is a class two graph with maximum degree Delta(G) that can be embedded on Sigma}. Hence Vizing's Planar Graph Conjecture can be restated as Delta(Sigma) = 5 if Sigma is a sphere. For a surface Sigma with Euler characteristic chi, it is known Delta(Sigma) >= H(chi) - 1 where H(chi) is the Heawood number of the surface and if the Euler characteristic chi is an element of {-7, -6, ..., -1, 0}, Delta(Sigma) is already known. In this paper, we study critical graphs on general surfaces and show that if G is a critical graph embeddable on a surface Sigma with Euler characteristic chi <= -8, then Delta(G) <= H(chi) (or H(chi) + 1) for some special families of graphs, namely if the minimum degree is at most 11 or if Delta is very large etc. As applications, we show that Delta(Sigma) <= H(chi) if chi is an element of (-22, -21, .., -8) \ {-19, -16 and Delta(Sigma) <= H(chi) + 1 if chi is an element of (-53, ..., -23} boolean OR (-19, -16}. Combining this with jungerman (1974), it follows that if chi = -12 and Sigma is orientable, then, Delta(Sigma) = H(chi). (C) 2019 Elsevier B.V. All rights reserved.