The Circular Chromatic Number

we start by defining the appropriate family of graphs. 

Definition 17.17. Let r he a real number, r 2. Then Rj. is defined to he the graph whose set of vertices is the set of unit vectors in the plane pointing from the origin, and two vertices x and y are connected hy an edge if and only if 27r/r a, where a is the sharper of the two angles between x and y (or -k if these two angles are equal). 

Note that both the number of vertices and valencies of the vertices (if r 2) are infinite. 

Definition 17.18. LetG be a graph. The circular chromatic number ofG 

Xc G) = inf r, where the infimum is taken over all positive reals r such that there exists a graph homomorphism from G to R.  

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We recommend an excellent and comprehensive textbook by Godsil and Royle, [GROl], where more about fractional chromatic number can be found. As for the circular chromatic number of a graph, we refer to nice articles [Vi88, ZhuOl] for rather extensive information. 

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