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.

翻译：$(n,m)$-图是一种包含$n$类有向弧和$m$类无向边的图。从$(n,m)$-图$G$到另一个$(n,m)$-图$H$的同态是一个保持邻接关系、方向及类型的顶点映射。使得$G$存在到$H$的同态的最小$|V(H)|$值称为$G$的$(n,m)$-染色数，记为$\mychi_{n,m}(G)$。该参数由Ne\v{s}et\v{r}il和Raspaud（J. Comb. Theory. Ser. B 2000）引入。本文证明：$G$的树状度受$\mychi_{n,m}(G)$的函数控制，但反之不成立；$G$的无圈染色数受$\mychi_{n,m}(G)$的函数控制，而反向界在已有文献中已知。进一步，对于最大平均度小于$2+ \frac{2}{4(2n+m)-1}$的图族（作为子族包含围长至少$8(2n+m)$的平面图族），其$(n,m)$-染色数等于$2(2n+m)+1$。这一结果改进了此前关于围长至少$10(2n+m)-4$的平面图族$(n,m)$-染色数为$2(2n+m)+1$的结论。已知部分$2$-树图族的$(n,m)$-染色数受$(2n+m)$的二次函数上下界控制，且在$(2n+m)=2$时下界是紧的。我们证明当$(2n+m)=3$时该下界不紧，并将对应下界改进了一个单位。同时，我们也改进了这些情况下的部分上界。