The fractional edge chromatic number of a graph 
is the fractional analog of the edge
chromatic number, denoted 
by Scheinerman and Ullman (2011). It can be defined
as
 |
(1)
|
where 
is the fractional chromatic number
of 
and 
is the line graph of 
.
There exists a polynomial-time algorithm for computing the fractional edge chromatic number (Scheinerman and Ullman 2011, pp. 86-87).
If the edge chromatic number of a graph equals its maximum vertex degree 
(i.e., if a graph is class
1), then the fractional edge chromatic number also equals 
. This follows from the general principle for fractional
objects that
 |
(2)
|
and the fact that
 |
(3)
|
(Scheinerman and Ullman 2011, p. 80), so combining gives
 |
(4)
|
Therefore, if 
, 
.
Since any vertex-transitive graph has either a perfect matching (for even vertex degree)
or a near-perfect matching (for odd vertex-degree; Godsil and Royle 2001, p. 43)
and every vertex-transitive graph has
its fractional chromatic number given
by the vertex count divided by its independence
number, applying the above to the line graph means
that a symmetric graph 
(i.e., one that is both vertex- and edge-transitive)
has fractional edge chromatic number given by
 |
(5)
|
where 
is the vertex count and 
the edge count of 
(S. Wagon, pers. comm., Jun. 6, 2012).
The flower snark 
is an example of a graph for which the edge
chromatic number 
and fractional edge chromatic number 
are both integers, but 
(Scheinerman and Ullman 2001, p. 96).
