On the chromatic polynomial of a graph

Discrete Applied Mathematics, 2008

In this note we consider a finite graph without loops and multiple edges. The colouring of a graph G in A colours is the colouring of its vertices in such a way that no two of adjacent vertices have the same colours and the number of used colours does not exceed A [1,4]. Two colourings of graph G are called different if there exists at least one vertex which changes colour when passing from one colouring to another. If F n is a full (also known as 'complete') graph on n vertices, & < n , then P(F k c F H , pi, A) = M (*>(A-*)<""*>, where M <*> = M (M-l)(/i-2). .. (/ *-* + 1). In particular P(F t <= F,,, A, A) = A W (A-*)<"-*> = A (n). If £ " is empty (also known as 'null' or 'totally disconnected' graph on n vertices) then P(E k a E, n fi, A) = fi k \"-k , in particular P(E k c £ " , A, A) = A". Let G = (X,V), \X\ = n be a graph and G o = (* " , K o), X 0 ^X 0 \X 0 \ = m be an induced subgraph of G. Let also x,y <= Xbe two non-adjacent vertices in G. We construct the graph G, from G by joining x and _y by an edge and the graph G 2 , obtained from G by contracting x and y into single vertex. Then we can observe that the following equality is true: P(G 0 ^G, ti, A) = P(G l 0 ^G u fi,\) + P(Gl<=G 2 , /i,A), (1) where Gi = Go = G o if x,_y ^ A'o, and G o , Go are the subgraphs induced respectively by X ih X 0 U{x,y} otherwise. It is so because P (G 0 c G 2 , M , A) equals the number of colourings for which x and y have the same colour. If we perform this operation further as much as possible we obtain P(G 0 = G, n, A) = X P{F kl c Fn,, M, A). /=i Since P(F k <= F n ,fj.,A) = P(F' k <^F n ,p,A) for any two fc-vertex complete subgraphs F A ., F' k of F,,, we can write P(G 0 cz G, n, A) = 5 t o, y P(F y ^ F h M , A). In order to clear up the meaning of coefficients a, y we consider one more class of colourings. A colouring of a graph is called an /-colouring, if exactly / colours are used, but Glasgow Math. J. 36 (1994) 265-267.

SIAM Journal on Discrete Mathematics, 2010

For a family S of graphs, let ω(S) be the supremum of t : 1 < t ≤ 2 such that P (G, λ) = 0 for all G ∈ S and all λ ∈ (1, t). In this paper we show that ω(S) = ω(S ∩ K) for any family S of graphs satisfying certain conditions, where K is a special family of graphs. This result makes it much easier to determine ω(S) for such families S.

Discrete Mathematics, 2015

If G is a k-chromatic graph of order n then it is known that the chromatic polynomial of G, π(G, x), is at most x(x − 1) • • • (x − (k − 1))x n−k = (x) ↓k x n−k for every x ∈ N. We improve here this bound by showing that π(G, x) ≤ (x) ↓k (x − 1) ∆(G)−k+1 x n−1−∆(G) for every x ∈ N, where ∆(G) is the maximum degree of G. Secondly, we show that if G is a connected k-chromatic graph of order n where k ≥ 4 then π(G, x) is at most (x) ↓k (x − 1) n−k for every real x ≥ n − 2 + n 2 − k 2 − n + k 2 (it had been previously conjectured that this inequality holds for all x ≥ k). Finally, we provide an upper bound on the moduli of the chromatic roots that is an improvment over known bounds for dense graphs.

Electronic Journal of Linear Algebra, 2012

In this paper, lower and upper bounds for the clique and independence numbers are established in terms of the eigenvalues of the signless Laplacian matrix of a given graph G.

arXiv (Cornell University), 2020

2017

Let $G$ be a graph of order $n$. It is well-known that $\alpha(G)\geq \sum_{i=1}^n \frac{1}{1+d_i}$, where $\alpha(G)$ is the independence number of $G$ and $d_1,\ldots,d_n$ is the degree sequence of $G$. We extend this result to digraphs by showing that if $D$ is a digraph with $n$ vertices, then $ \alpha(D)\geq \sum_{i=1}^n \left( \frac{1}{1+d_i^+} + \frac{1}{1+d_i^-} - \frac{1}{1+d_i}\right)$, where $\alpha(D)$ is the maximum size of an acyclic vertex set of $D$. Golowich proved that for any digraph $D$, $\chi(D)\leq \lceil \frac{4k}{5} \rceil+2$, where $k=max(\Delta^+(D),\Delta^-(D))$. We give a short and simple proof for this result. Next, we investigate the chromatic number of tournaments and determine the unique tournament such that for every integer $k&gt;1$, the number of proper $k$-colorings of that tournament is maximum among all strongly connected tournaments with the same number of vertices. Also, we find the chromatic polynomial of the strongly connected tournament wit...

The chromatic polynomials are studied by several authors and have important applications in di?erent frameworks, specially, in graph theory and enumerative combinatorics. The aim of this work is to establish some properties of the coe?fficients of the chromatic polynomial of a graph. Three applications on restricted Stirling numbers of the second kind are given.

European Journal of Combinatorics, 1990

Given a simple graph G = (V, E), a subset U of Vis called a clique if it induces a complete subgraph of G, and is called an independent set if it induces an empty subgraph of G. The cochromatic number z(G) of G is the minimum number of sets into which V can be partitioned so that each set is independent or a clique. Then z(G) is bounded above by the familiar chromatic number x(G) of G. Let w(G) denote the maximum cardinality among the cliques of G, and let n be an integer greater than 2. In this paper we explore the following question: Is there a function f(n) of n such that, if G has w(G)< n and is not the complete graph of order n-1, then x(G).s;z(G)+f(n)? Some bounds onf(n) are obtained and, in particular, it is shown that /(3) = 0 and /(4) = 1.

Discrete Mathematics, 2002

For a graph G, let P(G;) be its chromatic polynomial and let [G] be the set of graphs having P(G;) as their chromatic polynomial. We call [G] the chromatic equivalence class of G. If [G] = {G}, then G is said to be chromatically unique. In this paper, we ÿrst determine [G] for each graph G whose complement G is of the form aK1 ∪bK3 ∪ 16i6s P l i , where a; b are any nonnegative integers and li is even. By this result, we ÿnd that such a graph G is chromatically unique i ab = 0 and li = 4 for all i. This settles the conjecture that the complement of Pn is chromatically unique for each even n with n = 4. We also determine [H ] for each graph H whose complement H is of the form aK3 ∪ 16i6s Pu i ∪ 16j6t Cv j , where ui ¿ 3 and ui ≡ 4 (mod 5) for all i. We prove that such a graph H is chromatically unique if ui + 1 = vj for all i; j and ui is even when ui ¿ 6.