Chromatic number of signed graphs with bounded maximum degree

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