Pfaffian labelings and signs of edge colorings

Abstract. We relate signs of edge-colorings (as in classical Penrose's result) with "Pfaffian labelings", a generalization of Pfaffian orientations, whereby edges are labeled by elements of an Abelian group with an element of order two. In particular, we prove a conjecture of Goddyn that all k-edge-colorings of a k-regular Pfaffian graph G have the same sign. We characterize graphs that admit a Pfaffian labeling in terms of bricks and braces in their matching decomposition and in terms of their drawings in the projective plane.

• Type: Journal
• Year: 2008
• Where: COMBINATORICA
• Authors: Serguei Norine, Robin Thomas
• Keywords: Classical Penrose, COMBINATORICA 2008, Pfaffian, Pfaffian Labelings