Graph colorings and acyclic orientations

This paper presents new combinatorial proofs of two identities due to R. Stanley relating the chromatic polynomial and acyclic orientations of a graph. In addition, using elementary means, explicit formulae for the generating functions of the chromatic numbers and the number of color compatible acyclic orientations are derived. These formulae immediately show a reciprocity law concerning the generating functions.     

Graph decomposition with constraints on the connectivity and minimum degree

For each pair s,t of natural numbers there exist natural numbers f(s,t) and g(s,t) such that the vertex set of each graph of connectivity at least f(s,t) (respectively minimum degree at least g(s,t)) has a decomposition into sets which induce subgraphs of connectivity (respectively minimum degree) at least s and t, respectively.     

Graph maps whose periodic points form a closed set

A continuous map f from a graph G to itself is called a graph map. Denote by P(f), R(f), ω(f), Ω(f) and CR(f) the sets of periodic points, recurrent points, ω-limit points, non-wandering points and chain recurrent points of f respectively. It is well known that P(f)⊂R(f)⊂ω(f)⊂Ω(f)⊂CR(f). Block and Franke (1983) [5] proved that if f:I→I is an interval map and P(f) is a closed set, then CR(f)=P(f). In this paper we show that if f:G→G is a graph map and P(f) is a closed set, then ω(f)=R(f). We also give an example to show that, for general graph maps f with P(f) being a closed set, the conclusion ω(f)=R(f) cannot be strengthened to Ω(f)=R(f) or ω(f)=P(f).     

On the connectivity properties of the complementary set in fractal percolation models

We study the connectivity properties of the complementary set in Poisson multiscale percolation model and in Mandelbrot's percolation model in arbitrary dimension. By using a result about majorizing dependent random fields by Bernoulli fields, we prove that if the selection parameter is less than certain critical value, then, by choosing the scaling parameter large enough, we can assure that there is no percolation in the complementary set.     

Graphs with prescribed connectivity and line graph connectivity

Chartrand and Stewart have shown that the line graph of an n-connected graph is itself n-connected. This paper shows that for every pair of integers m > n > 1 there is a graph of point connectivity n whose line graph has point connectivity m. The corresponding question for line connectivity is also resolved.     

On the connectivity of the Julia set of a finitely generated rational semigroup

In this paper we show that the Julia set of a finitely generated rational semigroup is connected if the union of the Julia sets of generators is contained in a subcontinuum of . Under a nonseparating condition, we prove that the Julia set of a finitely generated polynomial semigroup is connected if its postcritical set is bounded.

     

Hypercube orientations with only two in-degrees

We consider the problem of orienting the edges of the n-dimensional hypercube so only two different in-degrees a and b occur. We show that this can be done, for two specified in-degrees, if and only if obvious necessary conditions hold. Namely, we need 0?a,b?n and also there exist non-negative integers s and t so that s+t=n² and as+bt=n²n−1. This is connected to a question arising from constructing a strategy for a “hat puzzle”.     

Two orientations

All trivializations of an Euclidean line bundle π: ? → B over a connected base B split in two classes which can be naturally named orientations of π. In the case of an orienting sheaf of a manifold or a vector bundle, they admit a natural interpretation as orientations of these objects. This approach establishes an extension of standard classical constructions to all manifolds and vector bundles independently of orientability restrictions in the usual sense.     

Graph Theory with Applications

Journal of the Operational Research Society -     

Minimum entropy orientations

We study graph orientations that minimize the entropy of the in-degree sequence. We prove that the minimum entropy orientation problem is NP-hard even if the graph is planar, and that there exists a simple linear-time algorithm that returns an approximate solution with an additive error guarantee of 1 bit.     

Graph drawings with few slopes

The slope-number of a graph G is the minimum number of distinct edge slopes in a straight-line drawing of G in the plane. We prove that for Δ5 and all large n, there is a Δ-regular n-vertex graph with slope-number at least . This is the best known lower bound on the slope-number of a graph with bounded degree. We prove upper and lower bounds on the slope-number of complete bipartite graphs. We prove a general upper bound on the slope-number of an arbitrary graph in terms of its bandwidth. It follows that the slope-number of interval graphs, cocomparability graphs, and AT-free graphs is at most a function of the maximum degree. We prove that graphs of bounded degree and bounded treewidth have slope-number at most . Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree. In a companion paper, planar drawings of graphs with few slopes are also considered.     

Graph with given achromatic number

An irreducible (point-determining) graph is one in which distinct vertices have distinct neighbourhoods. Every graph X can be reduced to an irreducible graph X¹ by identifying all vertices with the same neighbourhood; the colourability properties of X¹ carry over to X. Hence irreducible graphs are instrumental in the study of achromatic number. We prove that there are only finitely many irreducible graphs with a given achromatic number, and describe all graphs with achromatic number less than four. We deduce certain bounds on the achromatic number X in terms of the number of vertices of X¹. In the course of the proofs we calculate the achromatic numbers of paths and cycles. Generalizations of the main theorem to homomorphisms other than colourings are discussed.     

Graph coloring with decision diagrams

Mathematical Programming - We introduce an iterative framework for solving graph coloring problems using decision diagrams. The decision diagram compactly represents all possible color classes,...