  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.
