Grunbaum's Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A triangulation here is a cellular embedding of a finite graph in a closed surface with every face bounded by a walk of length three. Its dual $G^*$ has a vertex for each face and one edge crossing each primal edge. A proper $3$-edge-colouring assigns three colours to edges, with incident edges having different colours. Face-width is the least number of intersection points with the embedded graph among noncontractible simple closed curves on the surface. A cubic graph has degree three at every vertex; in the convention of this source, a snark is a cubic graph which is not $3$-edge-colourable and is internally $4$-edge-connected (no edge cut of size at most three separates two parts each having at least two vertices).

Statement recorded in the supplied source.

The original Grünbaum assertion says that if $G$ is a simple loopless triangulation of an orientable surface, then $G^*$ has a proper $3$-edge-colouring. The source gives its snark formulation: every orientable cellular embedding of a snark has a loop or parallel edges in the dual. It also records the following weaker questions: (i) is there an absolute constant $k$ such that the dual of every loopless orientable triangulation of face-width greater than $k$ is $3$-edge-colourable; (ii) does that assertion extend to nonorientable surfaces; and (iii) for every orientable surface $\Sigma$, is there a constant $k(\Sigma)$ giving this conclusion for loopless triangulations of $\Sigma$? For comparison, the separately named orientable cycle-double-cover conjecture in the source asks whether every snark has an orientable cellular embedding with a loopless dual. [S2]

Short English statement

The historical conjecture concerns three-edge-colourability of duals of simple orientable triangulations. The source also asks weaker versions with large face-width, either uniformly or depending on the surface, and discusses snark embeddings.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3314 (OPG-177). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Grunbaum's Conjecture, node 177, original problem page. Original question and full discussion. https://www.openproblemgarden.org/op/grunbaums_conjecture.