Title: Remarks on planar edge-chromatic critical graphs,
Abstract: 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.