Let
$G$
be a graph. The minimum number of colors needed to color the edges of
$G$
is called the chromatic index of
$G$
and is denoted by
$X'\left( G \right)$
. It is well known that
$\Delta \left( G \right)\,\le \,\mathcal{X}'\left( G \right)\,\le \Delta \left( G \right)\,+\,1$
, for any graph
$G$
, where
$\Delta \left( G \right)$
denotes the maximum degree of
$G$
. A graph
$G$
is said to be class 1 if
${\mathcal{X}}'\left( G \right)\,=\,\Delta \left( G \right)$
and class 2 if
${\mathcal{X}}'\left( G \right)\,=\,\Delta \left( G \right)\,+\,1$
. Also,
${{G}_{\Delta }}$
is the induced subgraph on all vertices of degree
$\Delta \left( G \right)$
. Let
$f:\,V\left( G \right)\,\to \mathbb{N}$
be a function. An
$f$
-coloring of a graph
$G$
is a coloring of the edges of
$E\left( G \right)$
such that each color appears at each vertex
$v\,\in \,V\left( G \right)$
at most
$f\left( v \right)$
times. The minimum number of colors needed to
$f$
-color
$G$
is called the
$f$
-chromatic index of
$G$
and is denoted by
${{{\mathcal{X}}'}_{f}}\left( G \right)$
. It was shown that for every graph
$G,\,{{\Delta }_{f}}\,\left( G \right)\,\le \,{{\mathcal{X}}^{\prime }}_{f}\left( G \right)\,\le \,{{\Delta }_{f}}\,\left( G \right)\,+\,1$
, where
${{\Delta }_{f}}\left( G \right)\,=\,{{\max }_{v\in \left( G \right)}}\,\left\lceil {{{d}_{G}}\left( v \right)}/{f\left( v \right)}\; \right\rceil $
. A graph
$G$
is said to be
$f$
-class 1 if
${{\mathcal{X}}^{\prime }}_{f}\left( G \right)\,=\,{{\Delta }_{f}}\left( G \right)$
, and
$f$
-class 2, otherwise. Also,
${{G}_{{{\Delta }_{f}}}}$
is the induced subgraph of
$G$
on
$\left\{ v\,\in \,V\left( G \right)\,:\,{{{d}_{G}}\left( V \right)}/{f\left( v \right)}\;\,=\,{{\Delta }_{f}}\left( G \right) \right\}$
. Hilton and Zhao showed that if
${{G}_{\Delta }}$
has maximum degree two and
$G$
is class 2, then
$G$
is critical,
${{G}_{\Delta }}$
is a disjoint union of cycles and
$\delta \left( G \right)\,=\,\Delta \left( G \right)-1$
, where
$\delta \left( G \right)$
denotes the minimum degree of
$G$
, respectively. In this paper, we generalize this theorem to
$f$
-coloring of graphs. Also, we determine the
$f$
-chromatic index of a connected graph
$G$
with
$\left| {{G}_{{{\Delta }_{f}}}} \right|\,\le \,4$
.