“Circular coloring”的意思、由来-开放百科全书

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Circular coloring

释义

Circular complete graphs
See also
References

In graph theory, circular coloring may be viewed as a refinement of usual graph coloring. The circular chromatic number of a graph , denoted can be given by any of the following definitions, all of which are equivalent (for finite graphs).

is the infimum over all real numbers so that there exists a map from to a circle of circumference 1 with the property that any two adjacent vertices map to points at distance along this circle.
is the infimum over all rational numbers so that there exists a map from to the cyclic group with the property that adjacent vertices map to elements at distance apart.
In an oriented graph, declare the imbalance of a cycle to be divided by the minimum of the number of edges directed clockwise and the number of edges directed counterclockwise. Define the imbalance of the oriented graph to be the maximum imbalance of a cycle. Now, is the minimum imbalance of an orientation of .

It is relatively easy to see that (especially using 1. or 2.), but in fact . 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 {{harvtxt|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.

Circular complete graphs

{{infobox graph
| name = Circular complete graph
| vertices = n
| edges = n(n-2k+1) / 2
| chromatic_number = ⌈n/k⌉
| girth =

| notation =

| properties = {{math|(n − 2k + 1)}}-regular
Vertex-transitive
Circulant
Hamiltonian
}}

For integers such that , the circular complete graph {{math|K_n/k}} (also known as a circular clique) is the graph with vertex set and edges between elements at distance apart.

That is, the vertices are numbers {0, 1, ..., n-1} and vertex i is adjacent to:

i+k, i+k+1, ..., i+n-k mod n.

For example, {{math|K_n/1}} is just the complete graph {{math|K_n}}, while {{math|K_{2n+1 / n}}} is isomorphic to the cycle graph {{math|C_2n+1}}.

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 {{math|K_a/b}} admits a homomorphism into {{math|K_c/d}} if and only if a/b ≤ c/d. This justifies the notation, since if the rational numbers a/b and c/d are equal, then {{math|K_a/b}} and {{math|K_c/d}} 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

{{math|K_2/1}} → {{math|K_5/2}} → {{math|K_7/3}} → ... → {{math|K_3/1}} → {{math|K_4/1}} → ...

or equivalently

{{math|K₂}} → {{math|C₅}} → {{math|C₇}} → ... → {{math|K₃}} → {{math|K₄}} → ...

The example on the figure can be interpreted as a homomorphism from the flower snark {{math|J₅}} into {{math|K_5/2 ≈ C₅}}, which comes earlier than {{math|K₃}}, corresponding to the fact that .

References

{{citation

| last = Nadolski | first = Adam
| contribution = Circular coloring of graphs
| doi = 10.1090/conm/352/09
| location = Providence, RI
| mr = 2076994
| pages = 123–137
| publisher = Amer. Math. Soc.
| series = Contemp. Math.
| title = Graph colorings
| volume = 352
| year = 2004}}.

{{citation

| last = Vince | first = A.
| doi = 10.1002/jgt.3190120411
| issue = 4
| journal = Journal of Graph Theory
| mr = 968751
| pages = 551–559
| title = Star chromatic number
| volume = 12
| year = 1988}}.

{{citation

| last = Zhu | first = X.
| doi = 10.1016/S0012-365X(00)00217-X
| issue = 1-3
| journal = Discrete Mathematics
| mr = 1815614
| pages = 371–410
| title = Circular chromatic number, a survey
| volume = 229
| year = 2001}}.{{combin-stub}}

3 : Graph coloring|Parametric families of graphs|Regular graphs

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Circular complete graphs

See also

References