Bounds on the chromatic polynomial and on the number of acyclic orientations of a graph

  title={Bounds on the chromatic polynomial and on the number of acyclic orientations of a graph},
  author={Nabil Kahal{\'e} and Leonard J. Schulman},
An upper bound is given on the number of acyclic orientations of a graph, in terms of the spectrum of its Laplacian. It is shown that this improves upon the previously known bound, which depended on the degree sequence of the graph. Estimates on the new bound are provided.A lower bound on the number of acyclic orientations of a graph is given, with the help of the probabilistic method. This argument can take advantage of structural properties of the graph: it is shown how to obtain stronger… 

Broken-Cycle-Free Subgraphs and the Log-Concavity Conjecture for Chromatic Polynomials

A computational verification of the strict log-concavity conjecture for chromatic polynomials for all graphs on at most 11 vertices, as well as for certain cubic graphs, is reported on.

Forests, colorings and acyclic orientations of the square lattice

Some asymptotic counting results about these quantities on then ×n section of the square lattice are obtained together with some properties of the structure of the random forest.

Upper bound for the number of spanning forests of regular graphs

In this paper, we study the number of spanning forests of regular graphs. Let τ(G) and F (G) denote the number of spanning trees and spanning forests, respectively. By a spanning forest F we mean an

Acyclic Orientations of Random Graphs

An acyclic orientation of an undirected graph is an orientation of its edges such that the resulting directed graph contains no cycles. The random graphG"n","pis a probability space consisting of

The Acyclic Orientation Game on Random Graphs

  • N. AlonZ. Tuza
  • Computer Science, Mathematics
    Random Struct. Algorithms
  • 1995
It is shown that in the random graph Gn p with (fixed) edge probability p > 0, the number of edges that have to be examined in order to identify an acyclic orientation is θ(n log n) almost surely.

Monomial bases for broken circuit complexes

On the Number of Upward Planar Orientations of Maximal Planar Graphs

It is shown that n-vertex maximal planar graphs have at least Ω(n·1.189 n ) and at most O(n ·4 n ) upward planar orientations.

A note on some inequalities for the Tutte polynomial of a matroid

Induced acyclic subgraphs in random digraphs: Improved bounds

A polynomial-time heuristic to find a large induced dag is analyzed and it is shown that it produces a solution whose size is at least $\log _q np + \Theta (\sqrt{\log_q np})$.



The effect of number of Hamiltonian paths on the complexity of a vertex-coloring problem

  • U. ManberM. Tompa
  • Mathematics
    22nd Annual Symposium on Foundations of Computer Science (sfcs 1981)
  • 1981
The proof of the lower bound involves showing that any dense graph must contain a subgraph with many Hamiltonian paths, and demonstrating the relevance of these Hamiltonianpaths to a geometric argument.

Information Bounds Are Weak in the Shortest Distance Problem

The characterization and enumeration of realizable patterns is studied, and it is shown thatf(n) 0 and the triangle inequalities d~j + dik > d,k, has at most C"' faces of all dimensions, thus resolving an open question in a similar information bound approach to the shortest distance problem.

Expander graphs

The lower bound on the expansion of Ramanujan graphs is improved, and a family of k-regular graphs with asymptotically optimal second eigenvalue and linear expansion equal to k/2 is constructed.

The Number of Spanning Trees in Regular Graphs

  • N. Alon
  • Mathematics
    Random Struct. Algorithms
  • 1990
It is shown that there is a function ϵ(k) that tends to zero as k tends to infinity such that for every connected, k-regular simple graph G on n vertices C(G) = {k[1 − δ(G)]}n.

Optimal randomized algorithms for local sorting and set-maxima

Lower bounds for the problems in the comparison model are described and it is shown that the algorithms are optimal within a constant factor.

Ramanujan graphs

The girth ofX is asymptotically ≧4/3 logk−1 ¦X¦ which gives larger girth than was previously known by explicit or non-explicit constructions.

Spanning Trees in Regular Graphs

Hard enumeration problems in geometry and combinatorics

A number of natural enumeration problems in geometry and combinatorics are shown to be complete in the class # P introduced by Valiant. Among others this is established for the numeration of vertices

The Probabilistic Method

A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.

A Randomised Approximation Algorithm for Counting the Number of Forests in Dense Graphs

  • J. Annan
  • Mathematics
    Combinatorics, Probability and Computing
  • 1994
The Tutte polynomial contains many other invariants of fundamental importance in fields as diverse as statistical physics, knot theory and graph colourings.