Circular coloring

In graph theory, circular coloring is a kind of coloring that may be viewed as a refinement of the usual graph coloring. The "circular chromatic number" of a graph formula_1, denoted formula_2 can be given by any of the following definitions, all of which are equivalent (for finite graphs).
It is relatively easy to see that formula_16 (especially using 1 or 2), but in fact formula_17. It is in this sense that we view circular chromatic number as a refinement of the usual chromatic number. 
Circular coloring was originally defined by , who called it "star coloring".
Coloring is dual to the subject of nowhere-zero flows and indeed, circular coloring has a natural dual notion: circular flows.
Circular complete graphs.
For integers formula_18 such that formula_19, the circular complete graph formula_20 (also known as a circular clique) is the graph with vertex set formula_21 and edges between elements at distance formula_22
That is vertex "i" is adjacent to:
formula_24 is just the complete graph , while formula_25 is the cycle graph formula_26
A circular coloring is then, according to the second definition above, a homomorphism into a circular complete graph. The crucial fact about these graphs is that formula_27 admits a homomorphism into formula_28 if and only if formula_29 This justifies the notation, since if formula_30 then formula_27 and formula_28 are homomorphically equivalent. Moreover, the homomorphism order among them refines the order given by complete graphs into a dense order, corresponding to rational numbers formula_33. For example
or equivalently
The example on the figure can be interpreted as a homomorphism from the flower snark into , which comes earlier than formula_36 corresponding to the fact that formula_37