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.
login/create account
Drupal
CSI of Charles University
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.