The f-Chromatic Index of Claw-free Graphs Whose f-Core is 2-regular

Let $G$ be a graph and $f:V(G)\rightarrow \mathbb{N}$ be a function. An $f$-coloring of a graph $G$ is an edge coloring 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 $χ'_{f}(G)$. It was shown that for every graph $G$, $Δ_{f}(G)\le χ'_{f}(G)\le Δ_{f}(G)+1$, where $Δ_{f}(G)=\max_{v\in V(G)} \lceil \frac{d_G(v)}{f(v)} \rceil$. A graph $G$ is said to be $f$-Class $1$ if $χ'_{f}(G)=Δ_{f}(G)$, and $f$-Class $2$, otherwise. Also, $G_{Δ_f}$ is the induced subgraph of $G$ on $\{v\in V(G):\,\frac{d_G(v)}{f(v)}=Δ_{f}(G)\}$. In this paper, we show that if $G$ is a connected graph with $Δ(G_{Δ_f})\leq 2$ and $G$ has an edge cut of size at most $Δ_f(G) -2$ which is a matching or a star, then $G$ is $f$-Class $1$. Also, we prove that if $G$ is a connected graph and every connected component of $G_{Δ_f}$ is a unicyclic graph or a tree and $G_{Δ_f}$ is not $2$-regular, then $G$ is $f$-Class $1$. Moreover, we show that except one graph, every connected claw-free graph $G$ whose $f$-core is $2$-regular with a vertex $v$ such that $f(v)\neq 1$ is $f$-Class $1$.

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