  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}
