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In previous papers the authors showed that almost all d-regular graphs for d≤3 are hamiltonian. In the present paper this result is generalized so that a set of j oriented root edges have been randomly specified for the cycle to contain. The Hamilton cycle must be orientable to agree with all of the orientations on the j root edges. It is shown that the requisite Hamilton cycle almost surely exists if and the limiting probability distribution at the threshold is determined when d=3. It is a corollary (in view of results elsewhere) that almost all claw-free cubic graphs are hamiltonian. There is a variation in which an additional cyclic ordering on the root edges is imposed which must also agree with their ordering on the Hamilton cycle. In this case, the required Hamilton cycle almost surely exists if j=o(n2/5). The method of analysis is small subgraph conditioning. This gives results on contiguity and the distribution of the number of Hamilton cycles which imply the facts above. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 128–147, 2001

We consider the problem of sampling independent sets of a graph with maximum degree δ. The weight of each independent set is expressed in terms of a fixed positive parameter λ≤2/(δ−2), where the weight of an independent set σ is λ|σ|. The Glauber dynamics is a simple Markov chain Monte Carlo method for sampling from this distribution. We show fast convergence (in O(n log n) time) of this dynamics. This paper gives the more interesting proof for triangle-free graphs. The proof for arbitrary graphs is given in a companion paper (E. Vigoda, Technical Report TR-99-003, International Computer Institute, Berkeley, CA, 1998). We also prove complementary hardness of approximation results, which show that it is hard to sample from this distribution when λ>c/δ for a constant c≤0. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 15, 229–241, 1999

The decycling number φ(G) of a graph G is the smallest number of vertices which can be removed from G so that the resultant graph contains no cycles. In this paper, we study the decycling numbers of random regular graphs. For a random cubic graph G of order n, we prove that φ(G) = ⌈n/4 + 1/2⌉ holds asymptotically almost surely. This is the result of executing a greedy algorithm for decycling G making use of a randomly chosen Hamilton cycle. For a general random d-regular graph G of order n, where d ≥ 4, we prove that φ(G)/n can be bounded below and above asymptotically almost surely by certain constants b(d) and B(d), depending solely on d, which are determined by solving, respectively, an algebraic equation and a system of differential equations.

In 1962 Posa conjectured that any graph G of order n and minimum degree at least ⅔ n contains the square of a Hamiltonian cycle. In this paper we prove this conjecture for sufficiently large n. © 1996 John Wiley & Sons, Inc.

This paper is an investigation of the structural properties of random plane-oriented recursive trees and their branches. We begin by an enumeration of these trees and some general properties related to the outdegrees of nodes. Using generalized Polya urn models we study the exact and limiting distributions of the size and the number of leaves in the branches of the tree. The exact distribution for the leaves in the branches is given by formulas involving second-order Eulerian numbers. A martingale central limit theorem for a linear combination of the number of leaves and the number of internal nodes is derived. The distribution of that linear combination is a mixture of normals with a beta distribution as its mixing density. The martingale central limit theorem allows easy determination of the limit laws governing the leaves in the branches. Furthermore, the asymptotic joint distribution of the number of nodes of outdegree 0, 1 and 2 is shown to be trivariate normal. © 1993 John Wiley & Sons, Inc.