Definition of fractional chromatic number

Suppose G  is a graph and r is a real number. Denote by I the collection of all independent sets of G. A fractional  r-coloring of G is an assignment f from I to R, which of non-negative weights to the independent sets of G such that the following hold:

         1)   For each vertex x,  sum_{x \in A, A \in I} f(A) = 1;

         2)   sum_{A \in I} f(A) <= r.

The fractional chromatic number, chi_f(G), of G is defined as

chi_f(G) = inf { r : G has a fractional r-coloring }.

An equivalent definition of the fractional chromatic number is as follows:

For an integer n, let [n] denote the set  {1, 2, ..., n }.  Given n > = 2k, the Kneser graph K(n, k)  has vertex set all the k-subsets of [n], in which two vertices X, X' are adjacent if and only if  X and X' do not intersect as subsets of [n].  A k-multiple n-coloring of G is a homomorphism from G to K(n, k). Or equivalently, a k-multiple n-coloring of G colors each vertex with k colors from the color set [n], and adjacent vertices are colored by disjoint color sets.  The fractional chromatic number of G is the infimum of the ratio n/k for which there is a k-multiple n-coloring of G.