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

can be given by any of the following definitions, all of which are equivalent (for finite graphs).

) ⌉ = χ (

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 Vince (1988), 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.

, the circular complete graph

(also known as a circular clique) is the graph with vertex set

and edges between elements at distance

That is vertex i is adjacent to:

is just the complete graph Kn, while

is the cycle graph

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

admits a homomorphism into

This justifies the notation, since if

are homomorphically equivalent.

Moreover, the homomorphism order among them refines the order given by complete graphs into a dense order, corresponding to rational numbers

For example or equivalently The example on the figure can be interpreted as a homomorphism from the flower snark J5 into K5/2 ≈ C5, which comes earlier than

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