Let 
be a fractional coloring of a graph 
. Then the sum of values of 
is called its weight, and the minimum possible weight of a
fractional coloring is called the fractional chromatic number 
, sometimes also denoted 
(Pirnazar and Ullman 2002, Scheinerman and Ullman 2011)
or 
(Larson et al. 1995), and sometimes also known as the set-chromatic number
(Bollobás and Thomassen 1979), ultimate chromatic number (Hell and Roberts
1982), or multicoloring number (Hilton et al. 1973). Every simple graph has
a fractional chromatic number which is a rational number or integer.
The fractional chromatic number of a graph can be obtained using linear programming, although the computation is NP-hard.
The fractional chromatic number of any tree and any bipartite graph is 2 (Pirnazar and Ullman 2002).
The fractional chromatic number satisfies
 |
(1)
|
where 
is the clique number, 
is the fractional
clique number, and 
is the chromatic number
(Godsil and Royle 2001, pp. 141 and 145), where the result 
follows from the strong duality theorem
for linear programming (Larson et al. 1995; Godsil and Royle 2001, p. 141).
The fractional chromatic number of a graph may be an integer that is less than the chromatic number. For example, for the Chvátal
graph, 
but 
.
Integer differences greater than one are also possible, for example, at least four
of the non-Cayley vertex-transitive graphs on 28 vertices have 
, and many Kneser graphs
have larger integer differences.
Gimbel et al. (2019) conjectured that every 4-chromatic planar graph has fractional chromatic number strictly greater than 3. Counterexamples
are provided by the 18-node Johnson skeleton
graph 
as well as the 18-node example given by Chiu et al. (2021) illustrated above.
Chiu et al. (2021) further demonstrated that there are exactly 17 4-regular
18-vertex planar graphs with chromatic number
4 and fractional chromatic number 3, and that there are no smaller graphs having
these values.
For any graph 
,
 |
(2)
|
where 
is the vertex count and 
is the independence
number of 
. Equality always holds for a vertex-transitive 
,
in which case
 |
(3)
|
(Scheinerman and Ullman 2011, p. 30). However, equality may also hold for graphs that are not vertex-transitive, including for the path
graph 
,
claw graph 
, diamond graph, etc.
Closed forms for the fractional chromatic number of special classes of graphs are given in the following table, where the Mycielski
graph 
is discussed by Larsen et al. (1995), the cycle graphs 
by Scheinerman and Ullman (2011, p. 31), and the
Kneser graph 
by Scheinerman and Ullman (2011, p. 32).
| graph | fractional chromatic number |
cycle graph  |  |
Kneser
graph 
for  |  |
Mycielski
graph  |  and  |
Other special cases are given in the following table.
| antiprism graph | 3, 4, 10/3, 3, 7/2, 16/5, 3, 10/3, 22/7, ... | |
| barbell graph | A000027 | 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ... |
cocktail party graph  | A000027 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ... |
complete graph  | A000027 | 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ... |
cycle graph  | A141310/A057979 | 3, 2, 5/2, 2, 7/3, 2, 9/4, 2, 11/5, 2, 13/6, ... |
empty
graph  | A000012 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... |
| helm graph | 4, 3, 7/2, 3, 10/3, 3, 13/4, 3, ... | |
Mycielski
graph  | A073833/A073834 | 2, 5/2, 29/10, 941/290, 969581/272890, ... |
| pan graph | A141310/A057979 | 3, 2, 5/2, 2, 7/3, 2, 9/4, 2, 11/5, 2, 13/6, ... |
prism
graph  | A141310/A057979 | 3, 2, 5/2, 2, 7/3, 2, 9/4, 2, 11/5, 2, 13/6, ... |
| sun graph | A000027 | 3, 4, 5, 6, 7, 8, 9, 10, 11, ... |
sunlet graph  | A141310/A057979 | 3, 2, 5/2, 2, 7/3, 2, 9/4, 2, 11/5, 2, 13/6, ... |
| web graph | 5/2, 2, 9/4, 2, 13/6, 2, 17/8, 2, 21/10, 2, 25/12, ... | |
wheel
graph  | 4, 3, 7/2, 3, 10/3, 3, 13/4, 3, 16/5, 3, 19/6, 3, ... |
-Colour Theorem for Planar Graphs." Bull. London
Math. Soc. 5, 302-306, 1973.Larsen, M.; Propp, J.; and Ullman,
D. "The Fractional Chromatic Number of Mycielski's Graphs." J. Graph
Th. 19, 411-416, 1995.Lovász, L. "Semidefinite
Programs and Combinatorial Optimization." In