The homomorphism and coloring of the direct product of signed graphs

Hedetniemi's conjecture, a significant problem in graph theory, was disproved by Shitov in 2019. This breakthrough stimulates us to consider the signed graphs version of Hedetniemi's conjecture, where signed graphs are graphs with an assignment of {±1} to the edges. In this paper, we provide counterexamples to the signed analog of Hedetniemi's conjecture. Nevertheless, we show that the chromatic number of the product is determined by its factors in some instances, i.e., if Γ→Θ and (Θ,π) is balanced, then χ((Γ,σ)×(Θ,π))=χ((Γ,σ)). As a byproduct, we investigate the homomorphisms from a factor of (Γ,σ)×(Θ,π) to the other, say from (Γ,σ) to (Θ,π). We find that if such homomorphisms exist, then the chromatic number of (Γ,+) provides a lower bound for χ((Γ,σ)×(Θ,π)). Moreover, if Γ is a core, then the number of balanced subgraphs isomorphic to (Γ,+) in (Γ,σ)×(Θ,π) is equal to the number of such homomorphisms.