Homomorphisms and edge-colourings of planar graphs

We conjecture that every planar graph of odd-girth 2 k + 1 admits a homomorphism to the Cayley graph C ( Z 2 2 k + 1 , S 2 k + 1 ) , with S 2 k + 1 being the set of ( 2 k + 1 ) -vectors with exactly two consecutive 1ʹs 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 ( 2 k + 1 ) -graph is ( 2 k + 1 ) -edge-colourable.