Homomorphism bounds of signed bipartite $K_4$-minor-free graphs and edge-colorings of $2k$-regular $K_4$-minor-free multigraphs

Abstract

A signed graph $(G,\Sigma )$ is a graph $G$ and a subset $\Sigma$ of its edges which corresponds to an assignment of signs to the edges: edges in $\Sigma$ are negative while edges not in $\Sigma$ 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 $(G,\Sigma )$ to $(H,\Pi )$ is a homomorphism of $G$ to $H$ which preserves the balance of closed walks. In this work, given a signed bipartite graph $(B,\Pi )$ of unbalanced-girth $2k$, we give a necessary and sufficient condition for $(B,\Pi )$ to admit a homomorphism from any signed bipartite graph of unbalanced-girth at least $2k$ whose underlying graph is $K_4$-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 ${\mathbb{Z}}_2^{2k-1}$ where vertices $u$ and $v$ are adjacent with a positive edge if their difference is in $\{e_1,e_2,\dots ,e_{2k-1}\}$ (where the $e_i$'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 $K_4$-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 $K_4$-minor-free multigraphs.

References in corpus (2)

• The complexity of signed graph and edge-coloured graph homomorphisms
• Homomorphism bounds and edge-colourings of $K_4$-minor-free graphs

Cited by in corpus (2)

• Complexity of planar signed graph homomorphisms to cycles
• Extended Double Covers and Homomorphism Bounds of Signed Graphs