On Chromatic Number of (n, m)-graphs

An (n, m)-graph is a graph with n types of arcs and m types of edges. The chromatic number of an (n, m)-graph G is the minimum number of colors to color the vertices of G such that if we identify the vertices of the same color we get a simple (n, m)-graph by identifying the arcs (resp., edges) between two vertices which have the same direction and color. We study this chromatic number for the family of sparse graphs, partial 2-trees when \(2n+m=3\) and for graphs with bounded maximum degree and degeneracy.