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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 .
When Birkhoff and Lewis introduced the chromatic polynomial in their attack on the four-color theorem, they conjectured that for planar graphs G, the polynomial (,) has no zeros in the region [,). Although it is known that such a chromatic polynomial has no zeros in the region [ 5 , ∞ ) {\displaystyle [5,\infty )} and that P ( G , 4 ) ≠ 0 ...
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
Bodlaender & Fomin (2005) showed that, given a graph G and a number c of colors, it is possible to test whether G admits an equitable c-coloring in time O(n O(t)), where t is the treewidth of G; in particular, equitable coloring may be solved optimally in polynomial time for trees (previously known due to Chen & Lih 1994) and outerplanar graphs ...
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 ...
Acyclic coloring; Chromatic polynomial; Cocoloring; Complete coloring; Edge coloring; Exact coloring; Four color theorem; Fractional coloring; Goldberg–Seymour ...
Pages in category "Graph coloring" The following 82 pages are in this category, out of 82 total. ... Chromatic polynomial; Circular coloring; Cocoloring; Col (game)