Martin Kochol and Bojan Mohar announced a counterexample to Grunbaum's conjecture at the PIMS Workshop on the Cycle Double Cover Conjecture (Vancouver, 2007). By using Kochol's "superposition" operation on several copies of Petersen's graph, they constructed a snark which embeds on the orientable surface of genus 9, and whose dual contains no loops or parallel edges.

Of course Grunbaum's Conjecture may still hold true for lower-genus surfaces, in particular, the torus.

## Grunbaum's conjecture is false!

Martin Kochol and Bojan Mohar announced a counterexample to Grunbaum's conjecture at the PIMS Workshop on the Cycle Double Cover Conjecture (Vancouver, 2007). By using Kochol's "superposition" operation on several copies of Petersen's graph, they constructed a snark which embeds on the orientable surface of genus 9, and whose dual contains no loops or parallel edges.

Of course Grunbaum's Conjecture may still hold true for lower-genus surfaces, in particular, the torus.

Ref: Kochol, M; Mohar, B; preprint 2007.