Andrew Vince

Star chromatic number

A generalization of the chromatic number of a graph is introduced such that the colors are integers modulo n, and the colors on adjacent vertices are required to be as far apart as possible.

Regular combinatorial maps

Abstract The classical approach to maps is by cell decomposition of a surface. A combinatorial map is a graph-theoretic generalization of a map on a surface. Besides maps on orientable and non-orientable surfaces, combinatorial maps include tessellations, hypermaps, higher dimensional analogues of maps, and certain toroidal complexes of Coxeter, Shephard, and Grunbaum. In a previous paper the incidence structure, diagram, and underlying topological space of a combinatorial map were investigated. This paper treats highly symmetric combinatorial maps. With regularity defined in terms of the automorphism group, necessary and sufficient conditions for a combinatorial map to be regular are given both graph- and group-theoretically. A classification of regular combinatorial maps on closed simply connected manifolds generalizes the well-known classification of metrically regular polytopes. On any closed manifold with nonzero Euler characteristic there are at most finitely many regular combinatorial maps. However, it is shown that, for nearly any diagram D , there are infinitely many regular combinatorial maps with diagram D . A necessary and sufficient condition for the regularity of rank 3 combinatorial maps is given in terms of Coxeter groups. This condition reveals the difficulty in classifying the regular maps on surfaces. In light of this difficulty an algorithm for generating a large class of regular combinatorial maps that are obtained as cyclic coverings of a given regular combinatorial map is given.

On representations of some thickness-two graphs

This paper considers representations of graphs as rectanglevisibility graphs and as doubly linear graphs. These are, respectively, graphs whose vertices are isothetic rectangles in the plane with adjacency determined by horizontal and vertical visibility, and graphs that can be drawn as the union of two straight-edged planar graphs. We prove that these graphs have, with n vertices, at most 6n−20 (resp., 6n−18) edges, and we provide examples of these graphs with 6n−20 edges for each n≥8.

The Hausdorff Dimension of the Boundary of a Self-Similar Tile

An effective method is given for computing the Hausdorff dimension of the boundary of a self-similar digit tile T in n -dimensional Euclidean space: formula here where 1/c is the contraction factor and λ is the largest eigenvalue of a certain contact matrix first defined by Grochenig and Haas.

The chaos game on a general iterated function system

The main theorem of this paper establishes conditions under which the ‘chaos game’ algorithm almost surely yields the attractor of an iterated function system. The theorem holds in a very general setting, even for non-contractive iterated function systems, and under weaker conditions on the random orbit of the chaos game than obtained previously.

Strongly balanced graphs and random graphs

The concept of strongly balanced graph is introduced. It is shown that there exists a strongly balanced graph with v vertices and e edges if and only if I ν – 1 ⩽ e ⩽(). This result is applied to a classic question of Erdos and Renyi: What is the probability that a random graph on n vertices contains a given graph? A rooted version of this problem is also solved.

A framework for the greedy algorithm

Perhaps the best known algorithm in combinatorial optimization is the greedy algorithm. A natural question is for which optimization problems does the greedy algorithm produce an optimal solution? In a sense this question is answered by a classical theorem in matroid theory due to Rado and Edmonds. In the matroid case, the greedy algorithm solves the optimization problem for every linear objective function. There are, however, optimization problems for which the greedy algorithm correctly solves the optimization problem for many--but not all--linear weight functions. Our intention is to put the greedy algorithm into a simple framework that includes such situations. For any pair (S,P) consisting of a finite set S together with a set P of partial orderings of S, we define the concepts of greedy set and admissible function. On a greedy set L ⊆ S, the greedy algorithm correctly solves the naturally associated optimization problem for all admissible functions f : S → R. Indeed, when P consists of linear orders, the greedy sets are characterized by this property. A geometric condition sufficient for a set to be greedy is given in terms of a polytope and roots that generalize Lie algebra root systems.

THE CONLEY ATTRACTORS OF AN ITERATED FUNCTION SYSTEM

We investigate the topological and metric properties of attractors of an iterated function system (IFS) whose functions may not be contractive. We focus, in particular, on invertible IFSs of nitely many maps on a com-pact metric space. We rely on ideas of Kieninger and McGehee and Wiandt, restricted to what is, in many ways, a simpler setting, but focused on a special type of attractor, namely point-fibred invariant sets. This allows us to give short proofs of some of the key ideas. 10.1017/S0004972713000348

The average order of a subtree of a tree

Let T be a tree all of whose internal vertices have degree at least three. In 1983 Jamison conjectured in JCT B that the average order of a subtree of T is at least half the order of T. In this paper a proof is provided. In addition, it is proved that the average order of a subtree of T is at most three quarters the order of T. Several open questions are stated.

Arithmetic and Fourier transform for the PYXIS multi-resolution digital Earth model

Abstract This paper investigates a multi-resolution digital Earth model called PYXIS, which was developed by PYXIS Innovation Inc. The PYXIS hexagonal grids employ an efficient hierarchical labeling scheme for addressing pixels. We provide a recursive definition of the PYXIS grids, a systematic approach to the labeling, an algorithm to add PYXIS labels, and a discussion of the discrete Fourier transform on PYXIS grids.