Finding the exact bound of the maximum degrees of class two graphs embeddable in a surface of characteristic ϵ∈{−1,−2,−3}

In this paper, we consider the problem of determining the maximum of the set of maximum degrees of class two graphs that can be embedded in a surface. For each surface Σ, we define Δ(Σ)=max{Δ(G)| G is a class two graph of maximum degree Δ that can be embedded in Σ}. Hence Vizing's Planar Graph Conjecture can be restated as Δ(Σ)=5 if Σ is a plane. We show that Δ(Σ)=7 if ϵ(Σ)=−1 and Δ(Σ)=8 if ϵ(Σ)∈{−2,−3}.