  In this paper, by developing several new adjacency lemmas about a path on $4$
or $5$ vertices, we show that the average degree of 7-critical graphs is at
least 6. It implies Vizing's planar graph conjecture for planar graphs with
maximum degree $7$ and its extension to graphs embeddable in a surface with
nonnegative Euler characteristic due to Sanders and Zhao (J. Combin. Theory
Ser. B 83 (2001) 201-212 and J. Combin. Theory Ser. B 87 (2003) 254-263) and
Zhang (Graphs and Combinatorics 16 (2000) 467-495).
