Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Remarks on planar edge-chromatic critical graphs   The only open case of Vizing's conjecture that every planar graph with
$\Delta\geq 6$ is a class 1 graph is $\Delta = 6$. We give a short proof of the
following statement: there is no 6-critical plane graph $G$, such that every
vertex of $G$ is incident to at most three 3-faces. A stronger statement
without restriction to critical graphs is stated in \cite{Wang_Xu_2013}.
However, the proof given there works only for critical graphs. Furthermore, we
show that every 5-critical plane graph has a 3-face which is adjacent to a
$k$-face $(k\in \{3,4\})$.
  For $\Delta = 5$ our result gives insights into the structure of planar
$5$-critical graphs, and the result for $\Delta=6$ gives support for the truth
of Vizing's planar graph conjecture.

18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph
23
The categories of this paper are: solv-int
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
26
The categories of this paper are: bayes-an
12
The categories of this paper are: nucl-th
30
The categories of this paper are: q-alg
3
The categories of this paper are: physics
24
The categories of this paper are: dg-ga
11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
13
The categories of this paper are: quant-ph
35
The categories of this paper are: mtrl-th