The chromatic number of 2-edge-colored and signed graphs of bounded maximum degree

Abstract: A 2-edge-colored graph or a signed graph is a simple graph with two types of edges. A homomorphism from a 2-edge-colored graph $G$ to a 2-edge-colored graph $H$ is a mapping $\varphi: V(G) \rightarrow V(H)$ that maps every edge in $G$ to an edge of the same type in $H$. Switching a vertex $v$ of a 2-edge-colored or signed graph corresponds to changing the type of each edge incident to $v$. There is a homomorphism from the signed graph $G$ to the signed graph $H$ if after switching some subset of the vertices of $G$ there is a 2-edge-colored homomorphism from $G$ to $H$. The chromatic number of a 2-edge-colored (resp. signed) graph $G$ is the order of a smallest 2-edge-colored (resp. signed) graph $H$ such that there is a homomorphism from $G$ to $H$. The chromatic number of a class of graph is the maximum of the chromatic numbers of the graphs in the class. We study the chromatic numbers of 2-edge-colored and signed graphs (connected and not necessarily connected) of a given bounded maximum degree. More precisely, we provide exact bounds for graphs of maximum degree 2. We then propose specific lower and upper bounds for graphs of maximum degree 3, 4, and 5. We finally propose general bounds for graphs of maximum degree $k$, for every $k$.