Random Graph's Hamiltonicity is Strongly Tied to its Minimum Degree

Yahav Alon,Michael Krivelevich

doi:10.37236/8339

Abstract

We show that the probability that a random graph $G\sim G(n,p)$ contains no Hamilton cycle is $(1+o(1))Pr(\delta (G) < 2)$ for all values of $p = p(n)$. We also prove an analogous result for perfect matchings.

Highlights

Introduction and main resultsHamilton cycles are a central topic in modern graph theory, a fact that extends to the field of random graphs as well, with numerous and diverse results regarding the appearance of Hamilton cycles in random graphs obtained over recent years.A classical result by Komlos and Szemeredi [9], and independently by Bollobas [2], states that a random graph G ∼ G(n, p), with np − ln n − ln ln n → ∞, is asymptotically almost surely Hamiltonian
We show that the probability that a random graph G ∼ G(n, p) contains no Hamilton cycle is (1 + o(1))P r(δ(G) < 2) for all values of p = p(n)
Similar to the result by Komlos and Szemeredi and of Bollobas about the threshold probability of Hamiltonicity, a very early work by Erdos and Renyi [5] shows that whenever np − ln n → ∞, the random graph G ∼ G(n, p) asymptotically almost surely contains a perfect matching

Summary

Introduction and main results

Hamilton cycles are a central topic in modern graph theory, a fact that extends to the field of random graphs as well, with numerous and diverse results regarding the appearance of Hamilton cycles in random graphs obtained over recent years. Similar to the result by Komlos and Szemeredi and of Bollobas about the threshold probability of Hamiltonicity, a very early work by Erdos and Renyi [5] shows that whenever np − ln n → ∞, the random graph G ∼ G(n, p) asymptotically almost surely contains a perfect matching. Similar to the connection between Hamiltonicity and minimum degree 2, this statement is true when replacing the electronic journal of combinatorics 27(1) (2020), #P1.30 the property of containing a perfect matching with that of containing no isolated vertices It was later proved by Bollobas and Thomason [3] that the hitting times of the two properties are asymptotically almost surely exactly equal to each other. EG(v): the set of edges in a graph G incident to the vertex v

Graph theory

Binomial coefficients and binomial distribution

Proof of main theorem

The dense case

Perfect matching