Remarks on planar edge-chromatic critical graphs

The only open case of Vizing's conjecture that every planar graph with Δ ? 6 is a class 1 graph is Δ = 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 Wang and 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 ? { 3 , 4 } ) .For Δ = 5 our result gives insights into the structure of planar 5-critical graphs, and the result for Δ = 6 gives support for the truth of Vizing's planar graph conjecture.