Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the neighbour sum distinguishing index of planar graphs   Let $c$ be a proper edge colouring of a graph $G=(V,E)$ with integers
$1,2,\ldots,k$. Then $k\geq \Delta(G)$, while by Vizing's theorem, no more than
$k=\Delta(G)+1$ is necessary for constructing such $c$. On the course of
investigating irregularities in graphs, it has been moreover conjectured that
only slightly larger $k$, i.e., $k=\Delta(G)+2$ enables enforcing additional
strong feature of $c$, namely that it attributes distinct sums of incident
colours to adjacent vertices in $G$ if only this graph has no isolated edges
and is not isomorphic to $C_5$. We prove the conjecture is valid for planar
graphs of sufficiently large maximum degree. In fact even stronger statement
holds, as the necessary number of colours stemming from the result of Vizing is
proved to be sufficient for this family of graphs. Specifically, our main
result states that every planar graph $G$ of maximum degree at least $28$ which
contains no isolated edges admits a proper edge colouring
$c:E\to\{1,2,\ldots,\Delta(G)+1\}$ such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni
v}c(e)$ for every edge $uv$ of $G$.

3
The categories of this paper are: physics
20
The categories of this paper are: acc-phys
10
The categories of this paper are: q-bio
17
The categories of this paper are: q-fin
30
The categories of this paper are: q-alg
36
The categories of this paper are: cmp-lg
31
The categories of this paper are: ao-sci
23
The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat
27
The categories of this paper are: comp-gas
28
The categories of this paper are: alg-geom
18
The categories of this paper are: econ
5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
21
The categories of this paper are: adap-org
37
The categories of this paper are: supr-con