Resource-sharing system scheduling and circular chromatic number

Circular chromatic number and scheduling for a resource-sharing system was studied. It was proved that the rate of an optimal strongly fair scheduling of a graph G is equal to the reciprocal of the circular chromatic number of G. The rate of an optimal weakly fair scheduling of G was proved to be equal to the reciprocal of the fractional chromatic number of g. In the study of circular chromatic number, the target graph for the interleaved p-color, q-tuple coloring and the target graph of Bondy and Hell defined (p, q)-coloring were found to be homomorphically equivalent.