Density of $3$-critical signed graphs

TL;DR

This paper establishes a lower bound on the number of edges in 3-critical signed graphs, linking it to colorability, and applies this to planar and projective-planar graphs with girth conditions, using homomorphism techniques.

Contribution

It introduces a new lower bound on edges for 3-critical signed graphs and connects this to circular colorability and homomorphism to a specific signed graph, extending understanding of signed graph colorings.

Findings

Every 3-critical signed graph on n vertices has at least (3n-1)/2 edges.

Signed planar or projective-planar graphs with girth at least 6 are circular 3-colorable.

The girth condition for projective-planar graphs is optimal.

Abstract

We say that a signed graph is $k$-critical if it is not $k$-colorable but every one of its proper subgraphs is $k$-colorable. Using the definition of colorability due to Naserasr, Wang, and Zhu that extends the notion of circular colorability, we prove that every $3$-critical signed graph on $n$ vertices has at least $\frac{3n-1}{2}$ edges, and that this bound is asymptotically tight. It follows that every signed planar or projective-planar graph of girth at least $6$ is (circular) $3$-colorable, and for the projective-planar case, this girth condition is best possible. To prove our main result, we reformulate it in terms of the existence of a homomorphism to the signed graph $C_{3}^*$, which is the positive triangle augmented with a negative loop on each vertex.