Extended Double Covers and Homomorphism Bounds of Signed Graphs

TL;DR

This paper explores the properties of Extended Double Covers of signed graphs, proposing a conjecture that extends the Four-Color Theorem, and investigates homomorphism bounds for specific subclasses of signed graphs.

Contribution

It introduces a new conjecture linking planarity and Extended Double Covers of signed graphs, and proves it for signed $K_4$-minor free graphs, advancing the theory of graph homomorphisms.

Findings

Proved the conjecture for signed $K_4$-minor free graphs.

Developed nearly optimal homomorphism bounds for subclasses with girth restrictions.

Introduced the concept of weighted signed graphs.

Abstract

A \emph{signed graph} $(G, \sigma)$ is a graph $G$ together with an assignment $\sigma:E(G) \rightarrow \{+,-\}$. The notion of homomorphisms of signed graphs is a relatively new development which allows to strengthen the connection between the theories of minors and colorings of graphs. Following this thread of thoughts, we investigate this connection through the notion of Extended Double Covers of signed graphs, which was recently introduced by Naserasr, Sopena and Zaslavsky. More precisely, we say that a signed graph $(B, \pi)$ is planar-complete if any planar signed graph $(G, \sigma)$ which verifies the conditions of a basic no-homomorphism lemma with respect to $(B,\pi)$ admits a homomorphism to $(B, \pi)$. Our conjecture then is that: if $(B, \pi)$ is a connected signed graph with no positive odd closed walk which is planar-complete, then its Extended Double Cover ${\rm EDC}(B,\pi)$ is also planar-complete. We observe that this conjecture largely extends the Four-Color Theorem and is strongly connected to a number of conjectures in extension of this famous theorem. A given (signed) graph $(B,\pi)$ \emph{bounds} a class of (signed) graphs if every (signed) graph in the class admits a homomorphism to $(B,\pi)$. In this work, and in support of our conjecture, we prove it for the subclass of signed $K_4$-minor free graphs. Inspired by this development, we then investigate the problem of finding optimal homomorphism bounds for subclasses of signed $K_4$-minor-free graphs with restrictions on their girth and we present nearly optimal solutions. Our work furthermore leads to the development of weighted signed graphs.