Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A proper total coloring distinguishing adjacent vertices by sums of some
  product graphs   In this article, we consider a proper total coloring distinguishes adjacent
vertices by sums, if every two adjacent vertices have different total sum of
colors of the edges incident to the vertex and the color of the vertex.
Pilsniak and Wozniak \cite{PW} first introduced this coloring and made a
conjecture that the minimal number of colors need to have a proper total
coloring distinguishes adjacent vertices by sums is less than or equal to the
maximum degree plus $3$. We study proper total colorings distinguishing
adjacent vertices by sums of some graphs and their products. We find that these
graphs satisfy the conjecture.

20
The categories of this paper are: acc-phys
32
The categories of this paper are: atom-ph
33
The categories of this paper are: plasm-ph
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th
22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
16
The categories of this paper are: stat
26
The categories of this paper are: bayes-an
0
The categories of this paper are: hep-ph
31
The categories of this paper are: ao-sci
4
The categories of this paper are: cond-mat
27
The categories of this paper are: comp-gas
6
The categories of this paper are: astro-ph