Hamilton Cycles in Random Graphs with a Fixed Degree Sequence

Colin Cooper,Alan Frieze,Michael Krivelevich

doi:10.1137/080741379

Abstract

Let $\mathbf{d}=d_1\leq d_2\leq\dots\leq d_n$ be a nondecreasing sequence of $n$ positive integers whose sum is even. Let $\mathcal{G}_{n,\mathbf{d}}$ denote the set of graphs with vertex set $[n]=\{1,2,\dots,n\}$ in which the degree of vertex $i$ is $d_i$. Let $G_{n,\mathbf{d}}$ be chosen uniformly at random from $\mathcal{G}_{n,\mathbf{d}}$. It will be apparent from section 4.3 that all of the sequences we are considering will be graphic. We give a condition on $\mathbf{d}$ under which we can show that whp $\mathcal{G}_{n,\mathbf{d}}$ is Hamiltonian. This condition is satisfied by graphs with exponential tails as well those with power law tails.

Highlights

Let d = d1 ≤ d2 ≤ · · · ≤ dn be a fixed non-decreasing sequence of n positive integers, whose sum is even
We know much about the structure of random regular graphs
In [7] we showed that degree sequences satisfying (1) are important

Summary

Introduction

(See Cooper and Frieze [6] for the connectivity properties of random digraphs with a fixed degree sequence). They have been used in the context of massive graph models of telephone networks and the WWW, Aiello, Chung and Lu [1]. In [7] we showed that degree sequences satisfying (1) are important We considered those with power law and exponential tails and showed that they satisfied the conditions of Theorem 1: Power Law Tails: For integer l ≥ 1 we let νl denote the number of vertices of degree l. Note that whp the degree sequence of Gn,p, p = c/n, c constant, satisfies this condition.

Configurations

Non-Hamiltonian Degree Sequences

Proof of Theorem 2

Expansion of GR

Posa’s Lemma and its consequences

From configurations to graphs