
The $f$Chromatic Index of a Graph Whose $f$Core has Maximum Degree $2$
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 $\chi'(G)$.
It is wellknown that $\Delta(G) \leq \chi'(G) \leq \Delta(G)+1$, for any
graph $G$, where $\Delta(G)$ denotes the maximum degree of $G$. A graph $G$ is said to be
Class $1$ if $\chi'(G) = \Delta(G)$ and Class $2$ if
$\chi'(G) = \Delta(G) + 1$. Also, $G_\Delta$ is the induced subgraph on all vertices of degree $\Delta(G)$.
Let $f:V(G)\rightarrow \mathbb{N}$ be a function.
An $f$coloring of a graph $G$ is a coloring of the edges
of $E(G)$ such that each color appears at each vertex $v\in V(G)$ at
most $f (v)$ times. The minimum number of colors needed
to $f$color $G$ is called the $f$chromatic index of $G$ and
is denoted by $\chi'_{f}(G)$. It was shown that for every graph $G$, $\Delta_{f}(G)\le \chi'_{f}(G)\le \Delta_{f}(G)+1$, where $\Delta_{f}(G)=\max_{v\in V(G)} \big\lceil \frac{d_G(v)}{f(v)}\big\rceil$. A graph $G$ is said to be $f$Class $1$ if $\chi'_{f}(G)=\Delta_{f}(G)$, and $f$Class $2$, otherwise. Also, $G_{\Delta_f}$ is the induced subgraph of $G$ on $\{v\in V(G):\,\frac{d_G(v)}{f(v)}=\Delta_{f}(G)\}$.
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 $\delta(G)=\Delta(G)1$, where $\delta(G)$ 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 $G_{\Delta_f}\le 4$.
Keywords:$f$coloring, $f$Core, $f$Class $1$ Categories:05C15, 05C38 