Bruce A. Reed

A critical point for random graphs with a given degree sequence

Given a sequence of nonnegative real numbers λ0, λ1… which sum to 1, we consider random graphs having approximately λi n vertices of degree i. Essentially, we show that if Σ i(i - 2)λi > 0, then such graphs almost surely have a giant component, while if Σ i(i -2)λ. < 0, then almost surely all components in such graphs are small. We can apply these results to Gn,p,Gn.M, and other well-known models of random graphs. There are also applications related to the chromatic number of sparse random graphs.

The Size of the Giant Component of a Random Graph with a Given Degree Sequence

Given a sequence of nonnegative real numbers λ0, λ1, … that sum to 1, we consider a random graph having approximately λin vertices of degree i. In [12] the authors essentially show that if ∑i(i−2)λi>0 then the graph a.s. has a giant component, while if ∑i(i−2)λi<0 then a.s. all components in the graph are small. In this paper we analyse the size of the giant component in the former case, and the structure of the graph formed by deleting that component. We determine e, λ′0, λ′1 … such that a.s. the giant component, C, has en+o(n) vertices, and the structure of the graph remaining after deleting C is basically that of a random graph with n′=n−∣C∣ vertices, and with λ′in′ of them of degree i.

Finding odd cycle transversals

We present an O(mn) algorithm to determine whether a graph G with m edges and n vertices has an odd cycle transversal of order at most k, for any fixed k. We also obtain an algorithm that determines, in the same time, whether a graph has a half integral packing of odd cycles of weight k.

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.

Further algorithmic aspects of the local lemma

Copyright

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.

A Bound on the Strong Chromatic Index of a Graph

We show that the strong chromatic index of a graph with maximum degree�; is at most (2��)�2, for some�>0. This answers a question of Erdo�s and Ne�et�il.

The height of a random binary search tree

Let <i>H<inf>n</inf></i> be the height of a random binary search tree on <i>n</i> nodes. We show that there exist constants α = 4.311… and β = 1.953… such that <b>E</b>(<i>H<inf>n</inf></i>) = α<i>ln n</i> − β<i>ln ln n</i> + <i>O</i>(1), We also show that <b>Var</b>(<i>H<inf>n</inf></i>) = <i>O</i>(1).

Vertex colouring edge partitions

A partition of the edges of a graph G into sets {S1,..., Sk} defines a multiset Xv for each vertex v where the multiplicity of i in Xv is the number of edges incident to v in Si We show that the edges of every graph can be partitioned into 4 sets such that the resultant multisets give a vertex colouring of G. In other words, for every edge (u, v) of G, Xu ≠ Xv. Furthermore, if G has minimum degree at least 1000, then there is a partition of E(G) into 3 sets such that the corresponding multisets yield a vertex colouring.

Excluding any graph as a minor allows a low tree-width 2-coloring

This article proves the conjecture of Thomas that, for every graph G, there is an integer k such that every graph with no minor isomorphic to G has a 2-coloring of either its vertices or its edges where each color induces a graph of tree-widlh at most k. Some generalizations are also proved.