Let G $G$ be a simple graph and χ ′ ( G ) $\chi ^{\prime} (G)$ be the chromatic index of G $G$ . We call G $G$ a Δ ${\rm{\Delta }}$ -critical graph if χ ′ ( G − e ) = χ ′ ( G ) − 1 = Δ $\chi ^{\prime} (G-e)=\chi ^{\prime} (G)-1={\rm{\Delta }}$ for every edge e $e$ of G $G$ , where Δ ${\rm{\Delta }}$ is maximum degree of G $G$ . Let e = x y $e=xy$ be an edge of Δ ${\rm{\Delta }}$ -critical graph G $G$ and φ $\varphi $ be an (proper) edge Δ ${\rm{\Delta }}$ -coloring of G − e $G-e$ . An e-fan is a sequence F e = ( x , e , y , e 1 , z 1 , … , e p , z p ) ${F}^{e}=(x,e,y,{e}_{1},{z}_{1},\ldots ,{e}_{p},{z}_{p})$ of alternating vertices and distinct edges such that edge e i ${e}_{i}$ is incident with x $x$ or y $y$ , z i ${z}_{i}$ is another endvertex of e i ${e}_{i}$ and φ ( e i ) $\varphi ({e}_{i})$ is missing at a vertex before z i ${z}_{i}$ for each i $i$ with 1 ≤ i ≤ p $1\le i\le p$ . In this paper, we prove that if min { d ( x ) , d ( y ) } ≤ Δ − 1 $\min \{d(x),d(y)\}\le {\rm{\Delta }}-1$ , where d ( x ) $d(x)$ and d ( y ) $d(y)$ denote the degrees of vertices x $x$ and y $y$ , respectively, then colors missing at different vertices of V ( F e ) $V({F}^{e})$ are distinct. Clearly, a Vizing fan is an e $e$ -fan with the restricting that all edges e i ${e}_{i}$ being incident with one fixed endvertex of edge e $e$ . This result gives a common generalization of several recently developed new results on multifan, double fan, Kierstead path of four vertices, and broom. By treating some colors of edges incident with vertices of low degrees as missing colors, Kostochka and Stiebitz introduced C $C$ -fan. In this paper, we also generalize the C $C$ -fan from centered at one vertex to one edge.