Graph invariants, homomorphisms, and the Tutte polynomial
نویسنده
چکیده
There are various ways to define the chromatic polynomial P (G; z) of a graph G. Perhaps the first that springs to mind is to define it to be the graph invariant P (G; k) with the property that when k is a positive integer P (G; k) is the number of colourings of the vertices of G with k or fewer colours such that adjacent vertices receive different colours. One then has to prove that P (G; k) is indeed a polynomial in k. This can be done for example by an inclusion-exclusion argument, or by establishing that P (G; k) satisfies a deletion-contraction recurrence and using induction. However, we shall take an alternative approach and define a polynomial P (G; z) by specifying its coefficients as graph invariants that count what are called colour-partitions of the vertex set of G. It immediately emerges that P (G; k) does indeed count the proper vertex k-colourings of G. A further aspect of this approach is that we choose a basis different to the usual basis {1, z, z, . . .} for polynomials in z. This basis, {1, z, z(z− 1), . . .}, has the advantage that we are able to calculate the chromatic polynomial very easily for many graphs, such as complete multipartite graphs. In this chapter we develop some of the many properties of the chromatic polynomial, which has received intensive study ever since Birkhoff introduced it in 1912 [2], perhaps with an analytic approach to 4CC in mind. Although such an approach has not led to such a proof of 4CC being found, study of the chromatic polynomial has led to many advances in graph theory that might not otherwise have ben made. In the context of this book, the chromatic polynomial played a significant role historically in Tutte’s elucidation of tension-flow duality. (In the next chapter we look at Tutte’s eponymous polynomial, introduced as simultaneous generalization of the chromatic and flow polynomials.) More about graph colourings can be found in e.g. [4, ch. V], [6, ch. 5], and more about the chromatic polynomial in e.g. [1, ch. 9] and [9]. We approach the chromatic polynomial via the key property that vertices of the same colour in a proper colouring of G form an independent (stable) set in G.
منابع مشابه
Homomorphisms and Polynomial Invariants of Graphs
This paper initiates a study of the connection between graph homomorphisms and the Tutte polynomial. This connection enables us to extend the study to other important polynomial invariants associated with graphs, and closely related to the Tutte polynomial. We then obtain applications of these relationships in several areas, including Abelian Groups and Statistical Physics. A new type of unique...
متن کاملOn the tutte polynomial of benzenoid chains
The Tutte polynomial of a graph G, T(G, x,y) is a polynomial in two variables defined for every undirected graph contains information about how the graph is connected. In this paper a simple formula for computing Tutte polynomial of a benzenoid chain is presented.
متن کاملGraph polynomials and Tutte-Grothendieck invariants: an application of elementary finite Fourier analysis
This paper is based on a series of talks given at the Patejdlovka Enumeration Workshop held in the Czech Republic in November 2007. The topics covered are as follows. The graph polynomial, Tutte-Grothendieck invariants, an overview of relevant elementary finite Fourier analysis, the Tutte polynomial of a graph as a Hamming weight enumerator of its set of tensions (or flows), description of a fa...
متن کاملThe arithmetic Tutte polynomials of the classical root systems
Many combinatorial and topological invariants of a hyperplane arrangement can be computed in terms of its Tutte polynomial. Similarly, many invariants of a hypertoric arrangement can be computed in terms of its arithmetic Tutte polynomial. We compute the arithmetic Tutte polynomials of the classical root systems An, Bn, Cn, and Dn with respect to their integer, root, and weight lattices. We do ...
متن کاملA Rewriting Approach to Graph Invariants
The Generic Diamond Lemma of the author is applied to the problem of classifying all graph invariants satisfying a contract–delete recursion (like that of the chromatic polynomial). As expected, the recursion for the Tutte polynomial is found, along with some more degenerate invariants. The purpose of this exercise is to demonstrate techniques for applying the diamond lemma to diagrammatic calc...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2013