Finding Δ(Σ) for a Surface Σ of Characteristic-4

For each surface Sigma, we define Delta(Sigma) = max{Delta(G)| G is a class two graph of maximum degree Delta(G) that can be embedded in Sigma}. Hence, Vizing's Planar Graph Conjecture can be restated as Delta(Sigma) = 5, if Sigma is a sphere. In this article, by applying some newly obtained adjacency lemmas, we show that Delta(Sigma) = 8 if Sigma is a surface of characteristic chi(Sigma) = -4. Until now, all known Delta(Sigma)s satisfy Delta(Sigma) = J(chi(Sigma)) = left perpendicular 3 + root 13-6 chi(Sigma) right perpendicular. This is the first case where Delta(Sigma) = J(chi(Sigma))-1. (C) 2015 Wiley Periodicals, Inc.