The chromatic number of signed graphs with bounded maximum average degree

A signed graph is a simple graph with two types of edges: positive and negative edges. Switching a vertex $v$ of a signed graph corresponds to changing the type of each edge incident to $v$. A homomorphism from a signed graph $G$ to another signed graph $H$ is a mapping $\varphi: V(G) \rightarrow V(H)$ such that, after switching some of the vertices of $G$, $\varphi$ maps every edge of $G$ to an edge of $H$ of the same type. The chromatic number $\chi_s(G)$ of a signed graph $G$ is the order of a smallest signed graph $H$ such that there is a homomorphism from $G$ to $H$. The maximum average degree $mad(G)$ of a graph $G$ is the maximum of the average degrees of all the subgraphs of $G$. We denote $\mathcal{M}_k$ the class of signed graphs with maximum average degree less than $k$ and $\mathcal{P}_g$ the class of planar signed graphs with girth at least $g$. We prove: $\chi_s(\mathcal{M}_{\frac{8}{3}}) = 4$ which implies $\chi_s(\mathcal{P}_{8}) = 4$, $\chi_s(\mathcal{P}_{7}) \le 5$, $\chi_s(\mathcal{M}_{\frac{17}{5}}) \le 10$ which implies $\chi_s(\mathcal{P}_{5}) \le 10$, $\chi_s(\mathcal{M}_{4-\frac{8}{q+3}}) \le q+1$ with $q$ a prime power congruent to 1 modulo 4.