Topological graph theory
Superposition and constructions of graphs without nowhere-zero k-flows
European Journal of Combinatorics
Complexity of approximation of 3-edge-coloring of graphs
Information Processing Letters
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A polyhedral embedding in a surface is one in which any two faces have boundaries that are either disjoint or simply connected. In a cubic (3-regular) graph this is equivalent to the dual being a simple graph. In 1968, Grunbaum conjectured that every cubic graph with a polyhedral embedding in an orientable surface is 3-edge-colorable. For the sphere, this is equivalent to the Four-Color Theorem, but we have disproved the conjecture in the general form. In this paper we extend this result and show that if we restrict our attention to a class of cubic graphs with a polyhedral embedding in an orientable surface, then the computational complexity of the 3-edge-coloring problem and its approximation does not improve.