Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the 1-2-3-conjecture   A k-edge-weighting of a graph G is a function w: E(G)->{1,2,...,k}. An
edge-weighting naturally induces a vertex coloring c, where for every vertex v
in V(G), c(v) is sum of weights of the edges that are adjacent to vertex v. If
the induced coloring c is a proper vertex coloring, then w is called a
vertex-coloring k-edge weighting (VCk-EW). Karonski et al. (J. Combin. Theory
Ser. B 91 (2004) 151-157) conjectured that every graph admits a VC3-EW. This
conjecture is known as 1-2-3-conjecture. In this paper, frst, we study the
vertex-coloring edge-weighting of the cartesian product of graphs. Among some
results, we prove that the 1-2-3-conjecture holds for some infinite classes of
graphs. Moreover, we explore some properties of a graph to admit a VC2-EW

12
The categories of this paper are: nucl-th
1
The categories of this paper are: cs
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
27
The categories of this paper are: comp-gas
33
The categories of this paper are: plasm-ph
34
The categories of this paper are: chem-ph
36
The categories of this paper are: cmp-lg
11
The categories of this paper are: hep-lat
17
The categories of this paper are: q-fin
6
The categories of this paper are: astro-ph