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Encyclopedia > Fractional coloring

Fractional coloring is a topic in a young branch of graph theory known as fractional graph theory. It differs from the traditional graph coloring in the sense that it assigns sets of colors instead of colors to elements. A diagram of a graph with 6 vertices and 7 edges. ... A 3-coloring suits this graph, but fewer colors would result in adjacent vertices of the same color. ...


A b-fold coloring of a graph G is a assignment of sets of size b to vertices of a graph such that adjacent vertices receive disjoint sets. An a:b-coloring is a b-fold coloring out of a available colors. The b-fold chromatic number χb(G) is the least a such that an a:b-coloring exists.


The fractional chromatic number χf(G) is defined to be

Note that the limit exists because χb(G) is subadditive, meaning χa+b(G) ≤ χa(G) + χb(G). A sequence { an }, n ≥ 1, is called subadditive if it satisfies the inequality for all m and n. ...


Some properties of χf(G):

and

Here n(G) is the order of G, α(G) the independence number and ω(G) the clique number. One major problem that has plagued graph theory since its inception is the lack of consistency in terminology. ... One major problem that has plagued graph theory since its inception is the lack of consistency in terminology. ... One major problem that has plagued graph theory since its inception is the lack of consistency in terminology. ...


References

  • Scheinerman, Edward R.; Ullman, Daniel H. (1997). Fractional graph theory. New York: Wiley-Interscience. ISBN 0-471-17864-0.

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Visualizing complex analytic functions using domain coloring (4509 words)
For the sake of example, our first coloring will be very simple: let the point w=4 be red, the point w=−1 blue, the point w=0 green, and the rest of the w plane white (see figure 1, where the axes are shown for reference only; they don't belong to the coloring).
Domain coloring perhaps doesn't do justice to all their properties, but let's look at a few things that can be seen.
With our use of colors to keep track of arg f (z), this is the same as saying that the colors sweep through the color gradient N − P times as we move around γ.
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