Homomorphisms and edge-colourings of planar graphs

We conjecture that every planar graph of odd-girth 2k + 1 admits a homomorphism to Cayley graph C(Z 2 , S2k+1), with S2k+1 being the set of (2k + 1)vectors with exactly two consecutive 1’s in a cyclic order. This is an strengthening of a conjecture of T. Marshall, J. Nešetřil and the author. Our main result is to show that this conjecture is equivalent to the corresponding case of a conjecture of P. Seymour, stating that every planar (2k+1)-graph is (2k+1)edge-colourable.