Colin McDiarmid

On colouring random graphs

Let ω n denote a random graph with vertex set {1, 2, …, n }, such that each edge is present with a prescribed probability p , independently of the presence or absence of any other edges. We show that the number of vertices in the largest complete subgraph of ω n is, with probability one,

Acyclic coloring of graphs

A vertex coloring of a graph G is called acyclic if no two adjacent vertices have the same color and there is no two-colored cycle in G. The acyclic chromatic number of G, denoted by A(G), is the least number of colors in an acyclic coloring of G. We show that if G has maximum degree d, then A(G) = 0(d4/3) as d → ∞. This settles a problem of Erdos who conjectured, in 1976, that A(G) = o(d2) as d → ∞. We also show that there are graphs G with maximum degree d for which A(G) = Ω(d4/3/(log d)1/3); and that the edges of any graph with maximum degree d can be colored by 0(d) colors so that no two adjacent edges have the same color and there is no two-colored cycle. All the proofs rely heavily on probabilistic arguments.

Algorithmic theory of random graphs

The theory of random graphs has been mainly concerned with structural properties, in particular the most likely values of various graph invariants – see Bollobas [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. In this paper we survey some of the results in this area.

Random planar graphs

We study various properties of the random planar graph Rn, drawn uniformly at random from the class Pn of all simple planar graphs on n labelled vertices. In particular, we show that the probability that Rn is connected is bounded away from 0 and from 1. We also show for example that each positive integer k, with high probability Rn has linearly many vertices of a given degree, in each embedding Rn has linearly many faces of a given size, and Rn has exponentially many automorphisms.

Channel assignment and weighted coloring

In cellular telephone networks, sets of radio channels (colors) must be assigned to transmitters (vertices) while avoiding interterence. Often, the transmitters are laid out like vertices of a triangular lattice in the plane. We investigated the corresponding weighted coloring problem of assigning sets of colors to vertices of the triangular lattice so that the sets of colors assigned to adjacent vertices are disjoint. We present a hardness result and an efficient algorithm yielding an approximate solution.

Small transversals in hypergraphs

For each positive integerk, we consider the setAk of all ordered pairs [a, b] such that in everyk-graph withn vertices andm edges some set of at mostam+bn vertices meets all the edges. We show that eachAk withk≥2 has infinitely many extreme points and conjecture that, for every positive ε, it has only finitely many extreme points [a, b] witha≥ε. With the extreme points ordered by the first coordinate, we identify the last two extreme points of everyAk, identify the last three extreme points ofA3, and describeA2 completely. A by-product of our arguments is a new algorithmic proof of Turáns theorem.

Vertex-Colouring Edge-Weightings

A weighting w of the edges of a graph G induces a colouring of the vertices of G where the colour of vertex v, denoted cv, is

ON INTEGER POINTS IN POLYHEDRA

{\sum\nolimits_{e \mathrel\backepsilon v} {w{\left( e \right)}} }