Acyclic orientations and chromatic generating functions
نویسنده
چکیده
Let P (k) be the chromatic polynomial of a graph with n ≥ 2 vertices and no isolated vertices, and let R(k) = P (k + 1)/k(k + 1). We show that the coefficients of the polynomial (1 − t) ∑ ∞ k=1 R(k)t are nonnegative and we give a combinatorial interpretation to R(k) when k is a nonpositive integer.
منابع مشابه
Sinks in Acyclic Orientations of Graphs
Greene and Zaslavsky proved that the number of acyclic orientations of a graph G with a unique sink at a given vertex is, up to sign, the linear coefficient of the chromatic polynomial. We give three proofs of this result using pure induction, noncommutative symmetric functions, and an algorithmic bijection. We also prove their result that if e=u0v0 is an edge of G then the number of acyclic or...
متن کاملA new approach to compute acyclic chromatic index of certain chemical structures
An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph $G$ denoted by $chi_a '(G)$ is the minimum number $k$ such that there is an acyclic edge coloring using $k$ colors. The maximum degree in $G$ denoted by $Delta(G)$, is the lower bound for $chi_a '(G)$. $P$-cuts introduced in this paper acts as a powerfu...
متن کاملA Quasisymmetric Function Generalization of the Chromatic Symmetric Function
The chromatic symmetric function XG of a graph G was introduced by Stanley. In this paper we introduce a quasisymmetric generalization X G called the k-chromatic quasisymmetric function of G and show that it is positive in the fundamental basis for the quasisymmetric functions. Following the specialization of XG to χG(λ), the chromatic polynomial, we also define a generalization χ k G(λ) and sh...
متن کاملAcyclic, Star and Oriented Colourings of Graph Subdivisions
Let G be a graph with chromatic number χ(G). A vertex colouring of G is acyclic if each bichromatic subgraph is a forest. A star colouring of G is an acyclic colouring in which each bichromatic subgraph is a star forest. Let χa(G) and χs(G) denote the acyclic and star chromatic numbers of G. This paper investigates acyclic and star colourings of subdivisions. Let G′ be the graph obtained from G...
متن کاملEnumeration of paths and cycles and e-coefficients of incomparability graphs
We prove that the number of Hamiltonian paths on the complement of an acyclic digraph is equal to the number of cycle covers. As an application, we obtain a new expansion of the chromatic symmetric function of incomparability graphs in terms of elementary symmetric functions. Analysis of some of the combinatorial implications of this expansion leads to three bijections involving acyclic orienta...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Discrete Mathematics
دوره 232 شماره
صفحات -
تاریخ انتشار 2001