Finding \(\Delta(\Sigma)\) for a surface \(\Sigma\) of characteristic \(-4\) (Q2833122)

A graph is edge \(k\)-colorable if there is an edge coloring of the graph with \(k\) colors. A finite simple graph of maximum degree \(d\) is class one if it is edge \(d\)-colorable. Otherwise Vizing's theorem guarantees that it is edge \((d+1)\)-colorable, in which case it is said to be class two. Vizing's conjecture called the planar graph conjecture claims that every planar graph of maximum degree at least six is class one. Therefore, the statement that every class two planar graph has maximum degree less than 6 is equivalent to the planar graph conjecture. In general, given a closed surface, we may try to determine the maximum of the maximum degrees of all simple class two graphs that can be embedded in the surface. The paper deals one such problem as described in the title.