Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fall-colorings and b-colorings of graph products   Given a proper coloring $f$ of a graph $G$, a b-vertex in $f$ is a vertex
that is adjacent to every color class but its own. It is a b-coloring if every
color class contains at least one b-vertex, and it is a fall-coloring if every
vertex is a b-vertex. The b-chromatic number of $G$ is the maximum integer
$b(G)$ for which $G$ has a b-coloring with $b(G)$ colors, while the
fall-chromatic number and the fall-acromatic number of $G$ are, respectively,
the minimum and maximum integers $f_1(G),f_2(G)$ for which $G$ has a
fall-coloring. In this article, we explore the concepts of b-homomorphisms and
Type II homomorphisms, which generalize the concepts of b-colorings and
fall-colorings, and present some meta-theorems concerning products of graphs.
As a result, we derive some previously known facts about these metrics on graph
products. We also give a negative answer to a question posed by Kaul and
Mitillos about fall-colorings of perfect graphs.

16
The categories of this paper are: stat
4
The categories of this paper are: cond-mat
29
The categories of this paper are: funct-an
12
The categories of this paper are: nucl-th
1
The categories of this paper are: cs
7
The categories of this paper are: hep-th
15
The categories of this paper are: nucl-ex
11
The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
5
The categories of this paper are: gr-qc
9
The categories of this paper are: nlin
14
The categories of this paper are: math-ph
17
The categories of this paper are: q-fin
8
The categories of this paper are: hep-ex
24
The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph