A $b$-fold coloring of a graph G is an assignment of sets of size $b$ to vertices of a graph such that adjacent vertices receive disjoint sets. An $a:b$-coloring is a $b$-fold coloring out of $a$ available colors. The $b$-fold chromatic number ${\displaystyle \chi _{b}(G)}$ is the least $a$ such that an $a:b$-coloring exists.

The fractional chromatic number ${\displaystyle \chi _{f}(G)}$ is defined to be

${\displaystyle \chi _{f}(G)=\lim _{b\to \infty }{\frac {\chi _{b}(G)}{b}}=\inf _{b}{\frac {\chi _{b}(G)}{b}}}$.

The fractional edge chromatic number $\chi'_{f}(G)$ of a graph $G$ is the fractional analog of the edge chromatic number. It can be defined as $\chi'_f(G)=\chi_f(L(G))$, where $L(G)$ is the line graph of $G$.

Similarly, we can define $a:b$-edge-coloring for graphs $G$. Now I am interseted whether there are existed results on $a:2$-edge-coloring. I just want to fix $b$ to be 2 and consider corresponding questions.

Thanks in advance!