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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 .
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.
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 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 ...
chromatic Having to do with coloring; see color. Chromatic graph theory is the theory of graph coloring. The chromatic number χ(G) is the minimum number of colors needed in a proper coloring of G. χ ′(G) is the chromatic index of G, the minimum number of colors needed in a proper edge coloring of G. choosable choosability
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})} . [ 15 ]
If such a k-coloring exists, then we refer to the smallest k needed in order to properly color our graph as the chromatic number, denoted by χ(G). [2] The number of proper k -colorings is a polynomial function of k called the chromatic polynomial of our graph G (by analogy with the chromatic polynomial of undirected graphs) and can be denoted ...