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!


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).

(You have to be careful about the fact that when you double every edge, you also multiply the maximum degree by two).

For instance, here is a result that you can obtain using this observation:

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 $a\le 2\Delta+1+1/\lceil g/2\rceil$

E. Steffen, A refinement of Vizing's theorem, Discrete Math. 218 (2000), 289-291.

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.


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