**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 χ_{b}(*G*) is the least *a* such that an *a*:*b*-coloring exists. The **fractional chromatic number** χ_{f}(*G*) is defined to be Note that the limit exists because χ_{b}(*G*) is *subadditive*, meaning χ_{a+b}(*G*) ≤ χ_{a}(*G*) + χ_{b}(*G*). A sequence { an }, n ≥ 1, is called subadditive if it satisfies the inequality for all m and n. ...
Some properties of χ_{f}(*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. |