An update on $(n,m)$-chromatic numbers

An $(n,m)$-graph is a graph with $n$ types of arcs and $m$ types of edges. A homomorphism of an $(n,m)$-graph $G$ to another $(n,m)$-graph $H$ is a vertex mapping that preserves adjacency, its direction, and its type. The minimum value of $|V(H)|$ such that $G$ admits a homomorphism to $H$ is the $(n,m)$-chromatic number of $G$, denoted by $\mychi_{n,m}(G)$. This parameter was introduced by Ne\v{s}et\v{r}il and Raspaud (J. Comb. Theory. Ser. B 2000). In this article, we show that the arboricity of $G$ is bounded by a function of $\mychi_{n,m}(G)$, but not the other way round. We also show that acyclic chromatic number of $G$ is bounded by a function of $\mychi_{n,m}(G)$, while the other way round bound was known beforehand. Moreover, we show that $(n,m)$-chromatic number for the family of graphs having maximum average degree less than $2+ \frac{2}{4(2n+m)-1}$, which contains the family of planar graphs having girth at least $8(2n+m)$ as a subfamily, is equal to $2(2n+m)+1$. This improves the previously known result which proved that the $(n,m)$-chromatic number for the family planar graphs having girth at least $10(2n+m)-4$ is equal to $2(2n+m)+1$. It is known that the $(n,m)$-chromatic number for the family of partial $2$-trees bounded below and above by quadratic functions of $(2n+m)$ and that the lower bound is tight when $(2n+m)=2$. We show that the lower bound is not tight when $(2n+m)=3$ by improving the corresponding lower bounds by one. We manage to improve some of the upper bounds in these cases as well.