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The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a function of the number of colors and was originally defined by George David Birkhoff to study the four color problem .
To compute the chromatic number and the chromatic polynomial, this procedure is used for every =, …,, impractical for all but the smallest input graphs. Using dynamic programming and a bound on the number of maximal independent sets , k -colorability can be decided in time and space O ( 2.4423 n ) {\displaystyle O(2.4423^{n})} . [ 13 ]
Important graph polynomials include: The characteristic polynomial, based on the graph's adjacency matrix. The chromatic polynomial, a polynomial whose values at integer arguments give the number of colorings of the graph with that many colors. The dichromatic polynomial, a 2-variable generalization of the chromatic polynomial
Finally, the third branch of algebraic graph theory concerns algebraic properties of invariants of graphs, and especially the chromatic polynomial, the Tutte polynomial and knot invariants. The chromatic polynomial of a graph, for example, counts the number of its proper vertex colorings .
The chromatic symmetric function is a symmetric function invariant of graphs studied in algebraic graph theory, a branch of mathematics. It is the weight generating function for proper graph colorings , and was originally introduced by Richard Stanley as a generalization of the chromatic polynomial of a graph.
The choosability (or list colorability or list chromatic number) ch(G) of a graph G is the least number k such that G is k-choosable. More generally, for a function f assigning a positive integer f ( v ) to each vertex v , a graph G is f -choosable (or f -list-colorable ) if it has a list coloring no matter how one assigns a list of f ( v ...
χ(G) (using the Greek letter chi) is the chromatic number of G and χ ′(G) is its chromatic index; see chromatic and coloring. child In a rooted tree, a child of a vertex v is a neighbor of v along an outgoing edge, one that is directed away from the root. chord chordal 1.
Chromatic number, the smallest number of colors for the vertices in a proper coloring; Chromatic index, the smallest number of colors for the edges in a proper edge coloring; Choosability (or list chromatic number), the least number k such that G is k-choosable; Independence number, the largest size of an independent set of vertices