Let f be a fractional coloring of a graph G. Then the sum of values of f is called its weight, and the minimum possible weight of a fractional coloring is called the fractional chromatic number χ^*(G), sometimes also denoted χ_f(G) (Pirnazar and Ullman 2002, Scheinerman and Ullman 2011) or χ_F(G) , and sometimes also known as the set-chromatic number, ultimate chromatic number, or multicoloring number . 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.
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