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A proper edge coloring of a simple graph $G$ from some lists
assigned to the edges of $G$ is of interest. A. Hilton and P.
Johnson (1990) considered a necessary condition for the list
coloring of a graph and called it Hall's condition. They
introduced the Hall index of a graph $G$, $h'(G)$, as the smallest
positive integer $m$ such that there exists a list coloring
whenever the lists are of length at least $m$ and Hall's condition
is satisfied. They characterized all graphs $G$ with $h'(G)=1$. In
this paper we characterize the graphs with Hall index 2.
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