3-Regular Non 3-Edge-Colorable Graphs with Polyhedral Embeddings in Orientable Surfaces

Authors:
Martin Kochol
Affiliations:
MÚ SAV, Štefánikova 49, 814 73 Bratislava 1, Slovakia and FPV ŽU,
Venue:
Graph Drawing
Year:
2009

Citing 4
Cited 1

Topological graph theory

Topological graph theory
Snarks without small cycles

Journal of Combinatorial Theory Series B
Superposition and constructions of graphs without nowhere-zero k-flows

European Journal of Combinatorics
Families of Dot-Product Snarks on Orientable Surfaces of Low Genus

Graphs and Combinatorics

Note: Complexity of 3-edge-coloring in the class of cubic graphs with a polyhedral embedding in an orientable surface

Discrete Applied Mathematics

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Abstract

The Four Color Theorem is equivalent with its dual form stating that each 2-edge-connected 3-regular planar graph is 3-edge-colorable. In 1968, Grünbaum conjectured that similar property holds true for any orientable surface, namely that each 3-regular graph with a polyhedral embedding in an orientable surface has a 3-edge-coloring. Note that an embedding of a graph in a surface is called polyhedral if its geometric dual has no multiple edges and loops. We present a negative solution of this conjecture, showing that for each orientable surface of genus at least 5, there exists a 3-regular non 3-edge-colorable graph with a polyhedral embedding in the surface.