# $2$-fold edge $b$-coloring of graphs

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.

The $$(a:2)$$-edge-coloring problem is equivalent to replacing each edge by two parallel edges. As there are many results on edge-coloring that take the multiplicity $$\mu$$ into account, they naturally give results on $$(a:2)$$-edge-coloring (or $$(a:b)$$-edge-coloring, more generally).
Steffen (2000): Every simple graph $$G$$ with finite girth (=size of a smallest cycle) $$g$$ and maximum degree $$\Delta$$ has an $$(a:2)$$-edge-coloring with $$a\le 2\Delta+1+1/\lceil g/2\rceil$$
I should add that there are hard problems related to $$(a:2)$$-edge coloring. The Berge-Fulkerson conjecture states that every cubic bridgeless graph has a (6:2)-edge-coloring. The conjecture was generalized by Seymour to 𝑟-graphs. Little is known about this, unfortunately.