Homomorphisms of planar signed graphs to signed projective cubes

Abstract

We conjecture that every signed graph of unbalanced girth 2g, whose underlying graph is bipartite and planar, admits a homomorphism to the signed projective cube of dimension 2g-1. Our main result is to show that for a given g, this conjecture is equivalent to the corresponding case (κ = 2g) of a conjecture of Seymour claiming that every planar κ-regular multigraph with no odd edge-cut of less than k edges is k-edge-colorable. To this end, we exhibit several properties of signed projective cubes and establish a folding lemma for planar even signed graphs. © 2013 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.