Bounding signed bipartite partial t-trees and application to edge-coloring

Abstract

Given a signed bipartite graph $(B, π)$ of negative girth $2k$, we present a necessary and sufficient condition for it to have the following property: each signed bipartite graph $(G, σ)$ whose negative girth is at least $2k$ and whose underlying graph has treewidth at most $t$ admits a homomorphism to $(B, π)$. Applying the result on the signed projective cube $SPC(2k-1)$, we conclude that every signed bipartite graph of negative girth at least $2k$ whose underlying graph is a partial 3-tree admits a homomorphism to $SPC(2k-1)$. For planar partial 3-trees, applying duality we conclude that if $G$ is a planar $2k$-regular multigraph whose dual has treewidth at most 3 and such that every edge-cut $(X, V\backslash X)$, where $|X|$ is odd, has size at least $2k$, then $G$ is $2k$-edge-colorable. This supports a conjecture of Seymour which, in full generality, largely extends Tait's reformulation of the four-color theorem, claiming that the fractional edge-chromatic number of a planar multigraph determines its edge-chromatic number. Finally, noting the contrast between fractional isomorphism and quantum isomorphism, where the former admits a polynomial time algorithm while the latter is proved to be undecidable, and observing the similarities of these notions to the subject of our study, we ask if there is an algorithm to decide if an input signed graph $\widehat{B}$ has the following property: if a signed planar graph $\widehat{G}$ does not map to $\widehat{B}$, it would be because a cycle in $\widehat{G}$ does not map to $\widehat{B}$. In other words, minimal planar graphs that do not map to $\widehat{B}$ are signed cycles.