On Chromatic Polynomial and Ordinomial

1 1 Introduction 2 1 Introduction Suppose Γ is a graph with |V (Γ)| = n. For λ a positive integer, let [λ] = {1, 2,. .. , λ} be a set of λ distinct colors. A λ-coloring of Γ is a mapping f from V (Γ) to [λ]. Whenever for every two adjacent vertices u and v, f (u) = f (v), we will call f a proper coloring of Γ; otherwise, improper. When a proper λ-coloring exists, we call Γ a λ-colorable graph. The chromatic number of Γ, denoted by χ(Γ), is defined as the smallest λ such that Γ is λ-colorable, and if that is the case, we call Γ a λ-chromatic graph. As we are only interested in proper colorings of graphs using [λ] as our color set, we will drop the term " proper " and the prefix " λ " from " proper λ-coloring " throughout this thesis, unless stated otherwise. Colorings f and g are considered distinct, if there exists v ∈ V (Γ), such that f (v) = g(v). The chromatic function of a graph, C(Γ; λ), is the number of distinct colorings of Γ. Theorem 1.1: C(Γ; λ) is a degree n monic polynomial of λ. Proof: Let r be a positive integer and m r (Γ) denote the number of distinct r-color-partitions; an r-color-partition of V (Γ) is a partition of vertices into r nonempty subsets, known as color-classes, such that no two vertices in a subset are adjacent. Clearly, for r greater than n, m r (Γ) = 0. For r less than or equal to n, we can color each r-color-partition in λ (r) = λ(λ − 1)(λ − 2) · · · (λ − r + 1) ways; Hence, as there are m r (Γ) such partitions, we have C(Γ; λ) = n r=1 m r (Γ) λ (r). It is clear that for every r, λ (r) is a polynomial of λ which implies C(Γ; λ) is also a polynomial of λ. Furthermore, as there is one color-partition of V (Γ) into n color-classes, the coefficient m n (Γ) of λ (n) (a polynomial of degree n that has the highest degree among λ (r)) is equal to 1. This proves that C(Γ; λ) is monic with degree n. From now on, we will refer to C(Γ; λ) as the chromatic polynomial of graph Γ. Furthermore, …