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.

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Correspondence to Serguei Norine.

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Partially supported by NSF grants 0200595 and 0354742.

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Norine, S., Thomas, R. Pfaffian labelings and signs of edge colorings. Combinatorica 28, 99–111 (2008). https://doi.org/10.1007/s00493-008-2231-2

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Mathematics Subject Classification (2000)

  • 05C70
  • 05C10
  • 05C15
  • 05C75