### Homomorphisms of planar signed graphs to signed projective cubes

*Reza Naserasr, Edita Rollová, Éric Sopena*

#### 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 (k=2g) of a conjecture of Seymour claiming that
every planar k-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.

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