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

1. along this circle.

2. apart.

3. In an oriented graph, declare the imbalance divided by the minimum of the number of edges directed clockwise and the number of edges directed counterclockwise. Define the imbalance.

. 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

, the circular complete graph (also known as a circular clique That is vertex i is adjacent to:

is just the complete graph is the cycle graph

A circular coloring is then, according to the second definition above, a homomorphism are homomorphically equivalent. Moreover, the homomorphism order among them refines the order given by complete graphs into a dense order. For example

or equivalently

The example on the figure can be interpreted as a homomorphism from the flower snark into

See also

• Rank coloring

References

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