Homomorphism bounds of signed bipartite [formula omitted]-minor-free graphs and edge-colorings of [formula omitted]-regular [formula omitted]-minor-free multigraphs

Abstract

A signed graph (G,Σ) is a graph G and a subset Σ of its edges which corresponds to an assignment of signs to the edges: edges in Σ are negative while edges not in Σ are positive. A closed walk of a signed graph is balanced if the product of the signs of its edges (repetitions included) is positive, and unbalanced otherwise. The unbalanced-girth of a signed graph is the length of a shortest unbalanced closed walk (if such a walk exists). A homomorphism of (G,Σ) to (H,Π) is a homomorphism of G to H which preserves the balance of closed walks.In this work, given a signed bipartite graph (B,Π) of unbalanced-girth 2k, we give a necessary and sufficient condition for (B,Π) to admit a homomorphism from any signed bipartite graph of unbalanced-girth at least 2k whose underlying graph is K4-minor-free. The condition can be checked in polynomial time with respect to the order of B.Let SPC(2k) be the signed bipartite graph on vertex set Z22k−1 where vertices u andv are adjacent with a positive edge if their difference is in {e1,e2,…,e2k−1} (where the ei’s form the standard basis), and adjacent with a negative edge if their difference is J (that is, the all-1 vector). As an application of our work, we prove that every signed bipartite K4-minor-free graph of unbalanced-girth 2k admits a homomorphism to SPC(2k). This supports a conjecture of Guenin claiming that every signed bipartite planar graph of unbalanced-girth 2k admits a homomorphism to SPC(2k) (this would be an extension of the four-color theorem).We also give an application of our work to edge-coloring 2k-regular K4-minor-free multigraphs.