Bounds on Chromatic Polynomials

2001

For a given graph G, let P (G,λ) be the chromatic polynomial of G, where λ is considered to be a real number. In this paper, we study the bounds for P (G,λ)/P (G,λ − 1) and P (G,λ)/P (G − x, λ), where x is a vertex in G, λ ≥ n and n is the number of vertices of G.

arXiv (Cornell University), 2018

A famous and wide-open problem, going back to at least the early 1970's, concerns the classification of chromatic polynomials of graphs. Toward this classification problem, one may ask for necessary inequalities among the coefficients of a chromatic polynomial, and we contribute such inequalities when a chromatic polynomial $\chi_G(n) = \chi^*_0 \binom {n+d} d + \chi^*_1 \binom {n+d-1} d + \dots + \chi^*_d \binom n d$ is written in terms of a binomial-coefficient basis. For example, we show that $\chi^*_{ j } \le \chi^*_{ d-j }$, for $0 \le j \le \frac{ d }{ 2 }$. Similar results hold for flow and tension polynomials enumerating either modular or integral nowhere-zero flows/tensions of a graph. Our theorems follow from connections among chromatic, flow, tension, and order polynomials, as well as Ehrhart polynomials of lattice polytopes that admit unimodular triangulations. Our results use Ehrhart inequalities due to Athanasiadis and Stapledon and are related to recent work by Hersh--Swartz and Breuer--Dall, where inequalities similar to some of ours were derived using algebraic-combinatorial methods.

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.

Mathematical Programming, 2002

Let P(G, λ) be the chromatic polynomial of a graph G with n vertices, independence number α and clique number ω. We show that for every λ ≥ n,

arXiv: Combinatorics, 2018

It is well known that for a graph $G=(V,E)$ of order $n$, its chromatic polynomial $P(G, x)$ can be expressed as $\sum\limits_{i=1}^n(-1)^{n-i}a_{i}x^i$, where $a_i$'s are non-negative integers. The number $\epsilon(G)=\sum\limits_{i=1}^n (n-i)a_i/\sum\limits_{i=1}^n a_i$ is the mean size of a broken-cycle-free spanning subgraph of $G$. Lundow and Markstr\"{o}m conjectured that $\epsilon(T_n) \epsilon(Q,x)$ holds for all real $x 0$ holds for all non-complete graphs $G$ of order $n$ and all real $x<0$. The last inequality is obtained by applying Whitney's broken-cycle theorem and Greene and Zaslavsky's interpretation on $a_1$ by special acyclic orientations.

Discret. Comput. Geom., 2021

A famous and wide-open problem, going back to at least the early 1970’s, concerns the classification of chromatic polynomials of graphs. Toward this classification problem, one may ask for necessary inequalities among the coefficients of a chromatic polynomial, and we contribute such inequalities when a chromatic polynomial χG(n) = χ ∗ 0 ( n+d d ) + χ∗ 1 ( n+d−1 d ) + · · ·+ χ∗ d ( n d ) is written in terms of a binomial-coefficient basis. For example, we show that χj ≤ χ ∗ d− j , for 0 ≤ j ≤ d 2 . Similar results hold for flow and tension polynomials enumerating either modular or integral nowhere-zero flows/tensions of a graph. Our theorems follow from connections among chromatic, flow, tension, and order polynomials, as well as Ehrhart polynomials of lattice polytopes that admit unimodular triangulations. Our results use Ehrhart inequalities due to Athanasiadis and Stapledon and are related to recent work by Hersh–Swartz and Breuer–Dall, where inequalities similar to some of ours ...

Journal of Combinatorial Theory, Series B, 2000

Let P(G, *) denote the chromatic polynomial of a graph G. It is proved in this paper that for every connected graph G of order n and real number * n, (*&2) n&1 P(G, *)&*(*&1) n&2 P(G, *&1) 0. By this result, the following conjecture proposed by Bartels and Welsh is proved: P(G, n)(P(G, n&1)) &1 >e for every graph G of order n.

arXiv (Cornell University), 2018

A famous and wide-open problem, going back to at least the early 1970's, concerns the classification of chromatic polynomials of graphs. Toward this classification problem, one may ask for necessary inequalities among the coefficients of a chromatic polynomial, and we contribute such inequalities when a chromatic polynomial χ G (n) = χ * 0 n+d d + χ * 1 n+d−1 d + • • • + χ * d n d is written in terms of a binomial-coefficient basis. For example, we show that χ * j ≤ χ * d− j , for 0 ≤ j ≤ d 2. Similar results hold for flow and tension polynomials enumerating either modular or integral nowhere-zero flows/tensions of a graph. Our theorems follow from connections among chromatic, flow, tension, and order polynomials, as well as Ehrhart polynomials of lattice polytopes that admit unimodular triangulations. Our results use Ehrhart inequalities due to Athanasiadis and Stapledon and are related to recent work by Hersh-Swartz and Breuer-Dall, where inequalities similar to some of ours were derived using algebraic-combinatorial methods.

Discrete Mathematics, 2008

It is well known that (−∞, 0) and (0, 1) are two maximal zero-free intervals for all chromatic polynomials. Jackson [A zero-free interval for chromatic polynomials of graphs, Combin. Probab. Comput. 2 (1993), 325-336] discovered that (1, 32 27 ] is another maximal zero-free interval for all chromatic polynomials. In this note, we show that (1, 32 27 ] is actually a maximal zero-free interval for the chromatic polynomials of bipartite planar graphs.

Linear Algebra and its Applications, 2020

The k-independence number of a graph is the maximum size of a set of vertices at pairwise distance greater than k. A graph is called k-partially walk-regular if the number of closed walks of a given length l ≤ k, rooted at a vertex v, only depends on l. In particular, a distance-regular graph is also k-partially walk-regular for any k. In this note, we introduce a new family of polynomials obtained from the spectrum of a graph. These polynomials, together with the interlacing technique, allow us to give tight spectral bounds on the k-independence number of a k-partially walk-regular graph. Together with some examples where the bounds are tight, we also show that the odd graph O with odd has no 1-perfect code.