Homomorphisms and edge-colourings of planar graphs

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We conjecture that every planar graph of odd-girth 2k + 1 admits a homomorphism to Cayley graph C(Z2k+1 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. Neˇsetˇril 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. 1 Homomorphisms Let G and H be graphs. A homomorphism f of G to H is an edge preserving mapping of V (G) to V (H). The theory of graph homomorphisms can be viewed as a generalization of the theory of graph colourings, as a k-colouring of a graph G is exactly a homomorphism of G to the complete graph Kk. The existence of a homomorphism of G to H is normally denoted by G → H. Homomorphism deﬁnes a quasiorder (a reﬂexive and transitive binary relation) on the set of graphs, by G H if and only if G → ...

Reza Naserasr

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