  In an edge-coloring of a cubic graph, an edge is poor or rich, if the set of
colors assigned to the edge and the four edges adjacent it, has exactly five or
exactly three distinct colors, respectively. An edge is normal in an
edge-coloring if it is rich or poor in this coloring. A normal
$k$-edge-coloring of a cubic graph is an edge-coloring with $k$ colors such
that each edge of the graph is normal. We denote by $\chi'_{N}(G)$ the smallest
$k$, for which $G$ admits a normal $k$-edge-coloring. Normal edge-colorings
were introduced by Jaeger in order to study his well-known Petersen Coloring
Conjecture. It is known that proving $\chi'_{N}(G)\leq 5$ for every bridgeless
cubic graph is equivalent to proving Petersen Coloring Conjecture. Moreover,
Jaeger was able to show that it implies classical conjectures like Cycle Double
Cover Conjecture and Berge-Fulkerson Conjecture. Recently, two of the authors
were able to show that any simple cubic graph admits a normal
$7$-edge-coloring, and this result is best possible. In the present paper, we
show that any claw-free bridgeless cubic graph, permutation snark, tree-like
snark admits a normal $6$-edge-coloring. Finally, we show that any bridgeless
cubic graph $G$ admits a $6$-edge-coloring such that at least $\frac{7}{9}\cdot
|E|$ edges of $G$ are normal.
