Chromatic number of signed graphs with bounded maximum degree

A signed graph $(G, Σ)$ is a graph positive and negative ($Σ$ denotes the set of negative edges). To re-sign a vertex $v$ of a signed graph $(G, Σ)$ is to switch the signs of the edges incident to $v$. If one can obtain $(G, Σ')$ by re-signing some vertices of $(G, Σ)$, then $(G, Σ) \equiv (G, Σ')$. A signed graphs $(G, Σ)$ admits an homomorphism to $(H, Λ)$ if there is a sign preserving vertex mapping from $(G,Σ')$ to $(H, Λ)$ for some $(G, Σ) \equiv (G, Σ')$. The signed chromatic number $χ_{s}( (G, Σ))$ of the signed graph $(G, Σ)$ is the minimum order (number of vertices) of a signed graph $(H, Λ)$ such that $(G, Σ)$ admits a homomorphism to $(H, Λ)$. For a family $\mathcal{F}$ of signed graphs $χ_{s}(\mathcal{F}) = \text{max}_{(G,Σ) \in \mathcal{F}} χ_{s}( (G, Σ))$. We prove $2^{Δ/2-1} \leq χ_s(\mathcal{G}_Δ) \leq (Δ-1)^2. 2^{(Δ-1)} +2$ for all $Δ\geq 3$ where $\mathcal{G}_Δ$ is the family of connected signed graphs with maximum degree $Δ$. \end{abstract}

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