Some inequalities on chromatic polynomials

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.

In this paper, we find a new phenomenon on chromatic polynomials of graphs. Let $\chi_G(t)=a_0t^n-a_1t^{n-1}+...(-1)^ra_rt^{n-r}$ be the chromatic polynomial of a simple graph $G$. For any $q,k\in \Bbb{Z}$ with $0\le k\le \min\{r, q+r+1\}$, we show that the partial binomial sum $\sum_{i=0}^{k}{q\choose i}a_{k-i}$ of $a_i$ is bounded above by ${m+q\choose k}$ and below by ${r+q\choose k}$, i.e., \[ {r+q\choose k}\le \sum_{i=0}^{k}{q\choose i}a_{k-i}\le {m+q\choose k}. \]

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,

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.

Discussiones Mathematicae Graph Theory, 2007

Let G be a graph of order n with clique number ω(G), chromatic number χ(G) and independence number α(G). We show that χ(G) ≤ n+ω+1−α 2. Moreover, χ(G) ≤ n+ω−α 2 , if either ω + α = n + 1 and G is not a split graph or α+ω = n−1 and G contains no induced K ω+3 −C 5 .

COMBINATORICA, 1998

We prove that the total chromatic number of any graph with maximum degree is at most plus an absolute constant. In particular, we show that for su ciently large, the total chromatic number of such a graph is at most + 10 26. The proof is probabilistic.

European Journal of Combinatorics, 2007

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.

2008

In this paper, we shall prove that if the domination number of G is at most 2, then P (G, ) is zero-free in the interval (1, ), where = 2 + 1 6 3 12 √ 93 − 108 − 1 6 3 12 √ 93 + 108 = 1.317672196 . . . , and P (G, ) = 0 for some graph G with domination number 2. We also show that if (G) v(G) − 2, then P (G, ) is zero-free in the interval (1, ), where = 5 3 + 1 6 3 12 √ 69 − 44 − 1 6 3 12 √ 69 + 44 = 1.430159709 . . . , and P (G, ) = 0 for some graph G with (G) = v(G) − 2.

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.