Geodesic
The fractional chromatic number of the direct product of graphs
Glasgow mathematical journal, Tome 44 (2002) no. 1, pp. 103-115

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This paper discusses the fractional chromatic number of the direct product of graphs. It is proved that if H is a circulant graph G^k_d, or a Kneser graph, or a direct sum of such graphs, then for any graph G, \chi_f{\hskip1}(G\times H{\hskip1}) = {\text min}\{\chi_f{\hskip1}(G), \chi_f{\hskip1}(H{\hskip1})\}. 
Zhu, Xuding. The fractional chromatic number of the direct product of graphs. Glasgow mathematical journal, Tome 44 (2002) no. 1, pp. 103-115. doi: 10.1017/S0017089502010066
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