Density of $C_{-4}$-critical signed graphs

TL;DR

This paper explores the properties of $C_{-4}$-critical signed graphs, establishing bounds on their structure, linking graph coloring to homomorphisms to $C_{-4}$, and applying these results to planar graphs and a conjecture in signed graph theory.

Contribution

It characterizes $C_{-4}$-critical signed graphs, proves bounds on their edges, and connects 4-coloring of graphs to homomorphisms to $C_{-4}$, extending understanding of signed graph coloring.

Findings

Most $C_{-4}$-critical signed graphs have at least $rac{4n}{3}$ edges.

All signed bipartite planar graphs with negative girth ≥ 8 map to $C_{-4}$.

Counterexample of girth 6 disproves the conjecture extending the 4-color theorem.

Abstract

A signed bipartite (simple) graph $(G, \sigma)$ is said to be $C_{-4}$-critical if it admits no homomorphism to $C_{-4}$ (a negative 4-cycle) but every proper subgraph of it does. In this work, first of all we show that the notion of 4-coloring of graphs and signed graphs is captured, through simple graph operations, by the notion of homomorphism to $C_{-4}$. In particular, the 4-color theorem is equivalent to: Given a planar graph $G$, the signed bipartite graph obtained from $G$ by replacing each edge with a negative path of length 2 maps to $C_{-4}$. We prove that, except for one particular signed bipartite graph on 7 vertices and 9 edges, any $C_{-4}$-critical signed graph on $n$ vertices must have at least $\lceil\frac{4n}{3}\rceil$ edges, and that this bound or $\lceil\frac{4n}{3}\rceil+1$ is attained for each value of $n\geq 9$. As an application, we conclude that all signed bipartite planar graphs of negative girth at least $8$ map to $C_{-4}$. Furthermore, we show that there exists an example of a signed bipartite planar graph of girth $6$ which does not map to $C_{-4}$, showing $8$ is the best possible and disproving a conjecture of Naserasr, Rollova and Sopena, in extension of the above mentioned restatement of the 4CT.