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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.
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 polynomial of a chordal graph is easy to compute. Find a perfect elimination ordering v 1, v 2, …, v n. Let N i equal the number of neighbors of v i that come after v i in that ordering. For instance, N n = 0.
An example graph, with the properties of being planar and being connected, and with order 6, size 7, diameter 3, girth 3, vertex connectivity 1, and degree sequence <3, 3, 3, 2, 2, 1> In graph theory , a graph property or graph invariant is a property of graphs that depends only on the abstract structure, not on graph representations such as ...
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 ...
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
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 ...