An $(n,m)$-graph is characterised by having $n$ types of arcs and $m$ types of edges. A homomorphism of an $(n,m)$-graph $G$ to an $(n,m)$-graph $H$, is a vertex mapping that preserves adjacency, direction, and type. The $(n,m)$-chromatic number of $G$, denoted by $\chi_{n,m}(G)$, is the minimum value of $|V(H)|$ such that there exists a homomorphism of $G$ to $H$. The theory of homomorphisms of $(n,m)$-graphs have connections with graph theoretic concepts like harmonious coloring, nowhere-zero flows; with other mathematical topics like binary predicate logic, Coxeter groups; and has application to the Query Evaluation Problem (QEP) in graph database. In this article, we show that the arboricity of $G$ is bounded by a function of $\chi_{n,m}(G)$ but not the other way around. Additionally, we show that the acyclic chromatic number of $G$ is bounded by a function of $\chi_{n,m}(G)$, a result already known in the reverse direction. Furthermore, we prove that the $(n,m)$-chromatic number for the family of graphs with a maximum average degree less than $2+ \frac{2}{4(2n+m)-1}$, including the subfamily of planar graphs with girth at least $8(2n+m)$, equals $2(2n+m)+1$. This improves upon previous findings, which proved the $(n,m)$-chromatic number for planar graphs with girth at least $10(2n+m)-4$ is $2(2n+m)+1$. It is established that the $(n,m)$-chromatic number for the family $\mathcal{T}_2$ of partial $2$-trees is both bounded below and above by quadratic functions of $(2n+m)$, with the lower bound being tight when $(2n+m)=2$. We prove $14 \leq \chi_{(0,3)}(\mathcal{T}_2) \leq 15$ and $14 \leq \chi_{(1,1)}(\mathcal{T}_2) \leq 21$ which improves both known lower bounds and the former upper bound. Moreover, for the latter upper bound, to the best of our knowledge we provide the first theoretical proof.

翻译：$(n,m)$-图由$n$种弧和$m$种边类型刻画。$(n,m)$-图$G$到$(n,m)$-图$H$的同态是保持邻接关系、方向性和类型的顶点映射。$G$的$(n,m)$-色数用$\chi_{n,m}(G)$表示，定义为存在$G$到$H$的同态时$|V(H)|$的最小值。$(n,m)$-图同态理论与和谐着色、无处为零流等图论概念，与二元谓词逻辑、Coxeter群等其他数学分支存在联系，并应用于图数据库中的查询评估问题（QEP）。本文中，我们证明$G$的树状度受$\chi_{n,m}(G)$的函数约束，反之不成立。此外，我们证明$G$的无环色数受$\chi_{n,m}(G)$的函数约束，该结果在反方向上已有证。进一步，我们证明最大平均度小于$2+ \frac{2}{4(2n+m)-1}$的图族（包括围长至少为$8(2n+m)$的平面图子族）的$(n,m)$-色数等于$2(2n+m)+1$。这改进了先前关于围长至少为$10(2n+m)-4$的平面图$(n,m)$-色数为$2(2n+m)+1$的结论。我们证实部分$2$-树族$\mathcal{T}_2$的$(n,m)$-色数的下界和上界均为$(2n+m)$的二次函数，且当$(2n+m)=2$时下界是紧的。我们证明$14 \leq \chi_{(0,3)}(\mathcal{T}_2) \leq 15$和$14 \leq \chi_{(1,1)}(\mathcal{T}_2) \leq 21$，这改进了两个已知下界及前一个上界。此外，对于后一个上界，据我们所知，我们首次提供了理论证明。