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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 ]
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.
R. M. Foster had already observed that the chromatic polynomial is one such function, and Tutte began to discover more, including a function f = t(G) counting the number of spanning trees of a graph (also see Kirchhoff's theorem).
The polynomial + + is the Tutte polynomial of the bull graph. The red line shows the intersection with the plane =, which is essentially equivalent to the chromatic polynomial. The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial.
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
Polynomials of degree one, two or three are respectively linear polynomials, quadratic polynomials and cubic polynomials. [8] For higher degrees, the specific names are not commonly used, although quartic polynomial (for degree four) and quintic polynomial (for degree five) are sometimes used. The names for the degrees may be applied to the ...
The chromatic number of any graph equals one more than the length of the longest path in an acyclic orientation chosen to minimize this path length. Acyclic orientations are also related to colorings through the chromatic polynomial , which counts both acyclic orientations and colorings.