  Let $G$ be a simple graph, and let $\Delta(G)$ and $\chi'(G)$ denote the
maximum degree and chromatic index of $G$, respectively. Vizing proved that
$\chi'(G)=\Delta(G)$ or $\Delta(G)+1$. We say $G$ is $\Delta$-critical if
$\chi'(G)=\Delta+1$ and $\chi'(H)<\chi'(G)$ for every proper subgraph $H$ of
$G$. In 1968, Vizing conjectured that if $G$ is a $\Delta$-critical graph, then
$G$ has a 2-factor. Let $G$ be an $n$-vertex $\Delta$-critical graph. It was
proved that if $\Delta(G)\ge n/2$, then $G$ has a 2-factor; and that if
$\Delta(G)\ge 2n/3+12$, then $G$ has a hamiltonian cycle, and thus a 2-factor.
It is well known that every 2-tough graph with at least three vertices has a
2-factor. We investigate the existence of a 2-factor in a $\Delta$-critical
graph under "moderate" given toughness and maximum degree conditions. In
particular, we show that if $G$ is an $n$-vertex $\Delta$-critical graph with
toughness at least 3/2 and with maximum degree at least $n/3$, then $G$ has a
2-factor. In addition, we develop new techniques in proving the existence of
2-factors in graphs.
