I'm searching for a reference to a particular alternate definition of the fractional chromatic number of graphs.

Let me review the most common definition and basic properties first.

Let $ G $ be a (finite simple) graph. For any natural number $ n $, we define the multi-chromatic number $ \chi_n(G) $ as the least $ m $ such that you can have a multi-coloring function $ c $ mapping the vertices of $ G $ to $ n $ element subsets of $ [m] $ such that the color of any two adjacent vertices have no intersection. We then define the fractional chromatic number of the graph as $ \chi^*(G) = \inf_n \chi_n(G)/n $.

It can be proved that this infimum is equal to the limit and equal to $ \chi_k(G)/k $ for some number $ k $, and so it is always rational as well. The fractional chromatic number of a graph is bounded from above by its chromatic number $ \chi(G) $ and from below by its clique number $ \omega(G) $.

Now the alternate definition I'm asking for.

I define the multi-chromatic number as $ \chi_n(G) = \chi(GK_n) $. Here $ GK_n $ is the graph you get by replacing each vertex of $ G $ by an $ n $-clique and each edge by $ n^2 $ edges linking each vertex from one corresponding clique to the other. I then define the fractional chromatic number the same as above.

(I use the notation $ GK_n $ because the graph is the lexicographical product of $ G $ and $ K_n $.)

It is easy to prove that this definition gives the same value for the multi-chromatic number as the usual definition. Indeed, if you have a coloring of $ GK_n $ with $ m $ colors, you get a natural $ n $-multi-coloring of $ G $ with $ m $ colors by defining the color of each vertex to the set of colors of the corresponding vertices in $ GK_n $, and this construction can be reversed as well.

While it's easy to see that the definition is equivalent, I'd like a reference from an existing book or article that gives this definition. This would show that people other than me find this definition natural.

I know there are other alternate definitions of the fractional chromatic number, namely there is one that gives it as the solution of a particular linear program, one using Kneser graphs, or one that uses graph powers. I'm not interested in those for this question.

I have searched in Pavol Hell, Jaroslav Nesetril, *Graphs and Homomorphisms*, and in Edward R. Scheinerman, Daniel H. Ullman, *Fractional Graph Theory*. Both of these talk about the fractional chromatic number in detail, but they don't give the definition I need.