Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Homomorphisms of planar $(m, n)$-colored-mixed graphs to planar targets   An $(m, n)$-colored-mixed graph $G=(V, A_1, A_2,\cdots, A_m, E_1, E_2,\cdots,
E_n)$ is a graph having $m$ colors of arcs and $n$ colors of edges. We do not
allow two arcs or edges to have the same endpoints. A homomorphism from an
$(m,n)$-colored-mixed graph $G$ to another $(m, n)$-colored-mixed graph $H$ is
a morphism $\varphi:V(G)\rightarrow V(H)$ such that each edge (resp. arc) of
$G$ is mapped to an edge (resp. arc) of $H$ of the same color (and
orientation). An $(m,n)$-colored-mixed graph $T$ is said to be $P_g^{(m,
n)}$-universal if every graph in $P_g^{(m, n)}$ (the planar $(m,
n)$-colored-mixed graphs with girth at least $g$) admits a homomorphism to $T$.
  We show that planar $P_g^{(m, n)}$-universal graphs do not exist for
$2m+n\ge3$ (and any value of $g$) and find a minimal (in the number vertices)
planar $P_g^{(m, n)}$-universal graphs in the other cases.

16
The categories of this paper are: stat
6
The categories of this paper are: astro-ph
26
The categories of this paper are: bayes-an
24
The categories of this paper are: dg-ga
18
The categories of this paper are: econ
7
The categories of this paper are: hep-th
30
The categories of this paper are: q-alg
29
The categories of this paper are: funct-an
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
14
The categories of this paper are: math-ph
23
The categories of this paper are: solv-int
31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph
37
The categories of this paper are: supr-con