The notion of homomorphism of signed graphs, introduced quite recently, provides better interplay with the notion of minor and is thus of high importance in graph coloring. A newer, but equivalent, definition of homomorphisms of signed graphs, proposed jointly by the second and third authors of this paper and Thomas Zaslavsky, leads to a basic no-homomorphism lemma. According to this definition, a signed graph $(G, \sigma)$ admits a homomorphism to a signed graph $(H, \pi)$ if there is a mapping $\phi$ from the vertices and edges of $G$ to the vertices and edges of $H$ (respectively) which preserves adjacencies, incidences, and signs of closed walks (i.e., the product of the sign of their edges). For $ij=00, 01, 10, 11$, let $g_{ij}(G,\sigma)$ be the length of a shortest nontrivial closed walk of $(G, \sigma)$ which is, positive and of even length for $ij=00$, positive and of odd length for $ij=01$, negative and of even length for $ij=10$, negative and of odd length for $ij=11$. For each $ij$, if there is no nontrivial closed walk of the corresponding type, we let $g_{ij}(G, \sigma)=\infty$. If $G$ is bipartite, then $g_{01}(G,\sigma)=g_{11}(G,\sigma)=\infty$. In this case, $g_{10}(G,\sigma)$ is certainly realized by a cycle of $G$, and it will be referred to as the \emph{unbalanced-girth} of $(G,\sigma)$.
It then follows that if $(G,\sigma)$ admits a homomorphism to $(H, \pi)$, then $g_{ij}(G, \sigma)\geq g_{ij}(H, \pi)$ for $ij \in \{00, 01,10,11\}$.
Studying the restriction of homomorphisms of signed graphs on sparse families, in this paper we first prove that for any given signed graph $(H, \pi)$, there exists a positive value of $\epsilon$ such that, if $G$ is a connected graph of maximum average degree less than $2+\epsilon$, and if $\sigma$ is a signature of $G$ such that $g_{ij}(G, \sigma)\geq g_{ij}(H, \pi)$ for all $ij \in \{00, 01,10,11\}$, then $(G, \sigma)$ admits a homomorphism to $(H, \pi)$.
For $(H, \pi)$ being the signed graph on $K_4$ with exactly one negative edge, we show that $\epsilon=\frac{4}{7}$ works and that this is the best possible value of $\epsilon$. For $(H, \pi)$ being the negative cycle of length $2g$, denoted $UC_{2g}$, we show that $\epsilon=\frac{1}{2g-1}$ works.
As a bipartite analogue of the Jaeger-Zhang conjecture, Naserasr, Sopena and Rollovà conjectured in [Homomorphisms of signed graphs, {\em J. Graph Theory} 79 (2015)] that every signed bipartite planar graph $(G,\sigma)$ satisfying $g_{ij}(G,\sigma)\geq 4g-2$ admits a homomorphism to $UC_{2g}$. We show that $4g-2$ cannot be strengthened, and, supporting the conjecture, we prove it for planar signed bipartite graphs $(G,\sigma)$ satisfying the weaker condition $g_{ij}(G,\sigma)\geq 8g-2$.
In the course of our work, we also provide a duality theorem to decide whether a 2-edge-colored graph admits a homomorphism to a certain class of 2-edge-colored signed graphs or not.
Hyperoctahedral Eulerian Idempotents, Hodge Decompositions, and Signed Graph Coloring Complexes