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 φ : V (G) → V (H) such that, after switching some of the vertices of G, φ maps every edge of G to an edge of H of the same type. The chromatic number χs(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 Mk the class of signed graphs with maximum average degree less than k and Pg the class of planar signed graphs of girth at least g. We prove: • χs(P7) ≤ 5, • χs(M 17 5 ) ≤ 10 which implies χs(P5) ≤ 10, • χs(M4− 8 q+3 ) ≤ q + 1 with q a prime power congruent to 1 modulo 4.