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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 .
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 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 chromatic polynomial includes more information about the colorability of G than does the chromatic number. Indeed, χ is the smallest positive integer that is not a zero of the chromatic polynomial χ(G) = min{k : P(G, k) > 0}.
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.
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 ...
For odd values of n, W n is a perfect graph with chromatic number 3: the vertices of the cycle can be given two colors, and the center vertex given a third color. For even n, W n has chromatic number 4, and (when n ≥ 6) is not perfect. W 7 is the only wheel graph that is a unit distance graph in the Euclidean plane. [4]
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 ...