Chromatic polynomial
The chromatic polynomial indicates the number of possible node colors with colors for a graph , i.e. H. the number of times that all nodes of the graph are colored, so that nodes connected by an edge have different colors.
Examples
The chromatic polynomial of a graph with isolated nodes is . Each of the nodes can take on one of the colors independently of the others .
The chromatic polynomial of a complete graph is
The color of the first node can always be chosen freely and there are still colors left for the coloring of the -th node .
properties
For every graph there is a number , so for all . This number is the chromatic number of the graph and indicates how many colors are required for a permissible node coloring.
At first it is not clear that there is a polynomial in at all , but this can be shown inductively, since the following applies for all edges : (where is the graph that is created by the edge contraction of e).
literature
- Martin Aigner : Combinatorial theory. Springer, 1979, ISBN 0-387-90376-3
- Swamy M., Thulasiraman K .: Graphs, Networks and Algorithms. Warrior Pub. Co., 1980, ISBN 0-471-03503-3
- William Thomas Tutte : Graph Theory. Addison-Wesley, 1984, ISBN 0-201-13520-5
- Herbert Wilf , "Algorithms and Complexity", Prentice-Hall, 1986
- Graham R. (Ed.), Grötschel M. (Ed.), László L. (Ed.): Handbook of Combinatorics. Vol. 1, Elsevier, 1995, ISBN 0-262-07170-3
Web links
- Eric W. Weisstein : Chromatic Polynomial . In: MathWorld (English).