Random graph's Hamiltonicity is strongly tied to its minimum degree

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.


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. A classical result by Komlós and Szemerédi [9], and independently by Bollobás [2], states that a random graph G ∼ G(n, p), with np − ln n − ln ln n → ∞, is asymptotically almost surely Hamiltonian. It should also be noted that if np − ln n − ln ln n → −∞ then asymptotically almost surely δ(G) ≤ 1, and thus G is not Hamiltonian. The same exact statement is true if one replaces the graph property G is Hamiltonian with the property δ(G) ≥ 2 . This indicates a possible connection between the two properties, an indication made explicit when considering a stronger result proved by Bollobás in [2] and by Ajtai, Komlós and Szemerédi in [1], regarding the hitting time of Hamiltonicity. Consider a random graph process, defined as a sequence of (random) graphs on n verticesG(σ) = where σ is an ordering of the edges of K n chosen randomly and uniformly from among all n 2 ! such orderings. Set G 0 (σ) to be a graph with no edges, and for all 1 ≤ i ≤ n 2 , G i (σ) is obtained by adding the i-th edge according to the order σ to G i−1 (σ). The hitting time of a monotone and non-empty graph property P, which we will denote as τ P (G(σ)), is a random variable equal to the index i for which G i (σ) ∈ P but G i−1 (σ) / ∈ P. Denote by H the property of Hamiltonicity, and by D 2 the property of having minimum degree at least two. The result states that asymptotically almost surely τ D 2 (G(σ)) = τ H (G(σ)). Indeed, the result about Hamiltonicity in G(n, p) can be derived directly from the hitting time result, thus making the hitting time result stronger. In addition, the hitting time result indicates an explicit connection between the minimum degree of a random graph and the existence of a Hamilton cycle. The random graph process asymptotically almost surely becomes Hamiltonian at the exact same moment it has minimum degree at least two. This shows that minimum degree less than two is typically the chief obstacle for Hamiltonicity. In light of this, it seems natural to ask whether the connection between the two properties can be expressed more explicitly. Furthermore, since the latter result revolves around the hitting time, one can ask whether this connection can also be observed in random graphs that are much more dense than the threshold density of 1 2 (ln n + ln ln n + ω(n)). A partial answer to these questions has been given by McDiarmid and Yolov in 2016 [11]. Based on a result by Hefetz, Krivelevich and Szabó [7], the authors proved the following: If p ≤ 1 2 is such that p·n ln ln ln ln n ln n·ln ln ln n → ∞, then the probability of G ∼ G(n, p) failing to contain a Hamilton cycle is at most (1 − p) n · exp O ln n ln ln ln n ln ln ln ln n . One can observe that the probability of G ∼ G(n, p) having δ(G) < 2 is of order Θ (np(1 − p) n ), and so this result gives an explicit bound on the ratio between the probabilities of the negations of these two properties: 1 ≤ P r(G is not Hamiltonian) P r(δ(G) < 2) ≤ exp O ln n ln ln ln n ln ln ln ln n This however only partially answers our question due to two gaps: first, it does not cover all of our target range of p(n); second, it is far from being tight.
Here we close both these gaps, by proving the following main result: By proving Theorem 1 we cover all possible values of p, as well as achieve the asymptotically tight ratio of 1 + o(1). Going forward, we utilize some of the approaches we use to prove Theorem 1 in order to sketch a proof of an analogous result regarding perfect matchings in a random graph. Assume for simplicity that n is even. Similar to the result by Komlós and Szemerédi and of Bollobás about the threshold probability of Hamiltonicity, a very early work by Erdős and Rényi [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 also true when replacing the property of containing a perfect matching with that of containing no isolated vertices. In fact, it was later proved by Bollobás and Thomason [3] that the hitting times of the two properties are asymptotically almost surely exactly equal to each other. Utilizing some similar approaches to those used in the proof of Theorem 1 we sketch a proof the following: Theorem 2. Let 0 ≤ p = p(n) ≤ 1, and let G ∼ G(n, p). Then P r(G contains no perfect matching) = (1 + o(1))P r(δ(G) = 0).
Once more this theorem covers the full range and provides a very tight ratio.
This paper is structured as follows: in Section 2 we list the notations and definitions to be used throughout the paper, as well as some auxiliary results needed for our proofs. In Section 3 we provide a proof of Theorem 1. In Section 4 we sketch a proof of Theorem 2. Portions of the proofs in Section 3 and Section 4 and several of the techniques we employ on our proofs are inspired by [10].

Notation, definitions and auxiliary results
In this section we provide several definitions and results to be used in the following sections. Throughout the paper, it is assumed that all logarithmic functions are in the natural base, unless explicitly stated otherwise. We suppress the rounding notation occasionally to simplify the presentation. The following standard graph theoretic notations will be used: • N G (U ) : the external neighbourhood of a vertex subset U in the graph G, i.e.
• e G (U ): the number of edges spanned in a graph G by a vertex subset U . This will sometimes be abbreviated as e(U ), when the identity of G is clear from the context.
• e G (U, W ): the number of edges of G between the two disjoint vertex sets U, W . This will sometimes be abbreviated as e(U, W ) when G is clear from the context.
• E G (v): the set of edges in a graph G incident to the vertex v.

Graph theory
Definition 2.1. Let G = (V, E) be a graph, and let α > 0 and k a positive integer. The graph G is a (k, α)-expander if |N G (W )| ≥ α|W | for every vertex subset W ⊂ V , |W | ≤ k.

General auxiliary results
Lemma 2.9. (Particular case of Bonferroni inequality) Let {A k } n k=1 be a family of events in a probability space. Then:

Proof of main theorem
In this section we provide a proof for Theorem 1. The theorem's statement covers all possible values of the edge probability p(n), with the corresponding random graphs G(n, p) having somewhat different characteristics in different parts of this range. Since this is the case, we divide our proof into four parts, each corresponding to a different range of p. The general approach in each of the proofs, bar the first part, will be as thus: we will present a finite set of graph properties, say, {P i } i≥0 , with P 0 being the property of having minimum degree at least 2, and show that: (i) For a random graph G ∼ G(n, p) and for all i > 0, the probability of G / ∈ P i is o (P r(G / ∈ P 0 )); (ii) Any graph G such that G ∈ i≥0 P i is Hamiltonian.
Since a Hamiltonian graph G must have a minimum degree at least 2, meaning P r(δ(G) < 2) ≤ P r(G is not Hamiltonian), combining the two claims yields the theorem. The four parts of the proof will be the following: • The very sparse case: let p = p(n) be such that np − log n − log log n does not tend to infinity. In this case there is nothing to prove, since the theorem's statement is already known to be true. This is due to the result by Komlós and Szemerédi, stating that for G ∼ G(n, p): It follows that lim n→∞ Pr(G is not Hamiltonian) = lim n→∞ Pr(δ(G) < 2) = C > 0, and the statement therefore holds.
• The sparse case: in Section 3.1 we provide the proof of the theorem, assuming np − log n − log log n → ∞ and p ≤ 100 log n n . In the proof of this case, we aim to show that with the appropriate probability for G ∼ G(n, p), G contains an n 4 , 2 -expander as a subgraph that is sparse, relative to the average degree of G. Then invoking Lemma 2.3 we conclude that this subgraph can be made Hamiltonian with high probability by adding boosters that are also edges of G. In this range of p, our main concern is the existence of many vertices of degree sublinear in np. This concern is addressed by showing that with high probability these vertices are relatively few and far apart. This proof, up to the use of the G(n, p) model and the tightness of the bounds sought, is essentially due to [10].
• The dense case: in Section 3.2 we cover the case 100 log n n ≤ p ≤ 0.01. The proof of this case is fairly similar to the proof of the sparse case, and in fact somewhat simpler due to the fact that it is now highly unlikely for G to contain more than one vertex of degree sublinear in np.
• The very dense case: in Section 3.3 we provide the proof of the theorem, assuming p ≥ 0.01.
In this range the graph is indeed typically very dense, and so removing a minimum degree vertex will typically result in a graph whose independence number is smaller than its vertex connectivity. This fact allows for the use of the Chvátal-Erdős Theorem to prove Hamiltonicity of the original graph.

The sparse case
Recall that we assume here that np − log n − log log n → ∞ and p ≤ 100 log n n . Define and for a graph G = (V, E), denote Now, define the following graph properties: log n : e(U, W ) ≥ 1 2 n; (P7) Every n 4 , 2 -expanding subgraph Γ of G with at most (d 0 + 1)n edges is either Hamiltonian, or has a booster in E(G) \ E(Γ).
where ω(n) is such that lim n→∞ ω(n) = ∞ and p ≤ 100 log n n , and let G ∼ G(n, p) be a random graph. Then the probability that all properties Proof. Clearly, the probability that exists a property among (P0)-(P7) that does not hold is at least P r(G / ∈ (P0)), and at most 7 i=0 P r(G / ∈ (Pi)). We will prove that for all 1 ≤ i ≤ 7 one has P r(G / ∈ (Pi)) = o(P r(G / ∈ (P0))), and thus will establish the lemma. First, we bound P r(G / ∈ (P0)) from below: For a vertex v ∈ V (G), denote by A v the event "d(v) < 2". So P r(G / ∈ (P0)) = P r v∈V (G) A v . By Bonferroni's inequality (Lemma 2.9): Let u, v ∈ V (G). We first observe that Since log n < np ≤ 100 log n, we get: Now, the probability of A u ∩ A v is at most the probability that e G ({u, v}, V \ {u, v}) ≤ 2: . And so overall we get: For the rest of the properties we bound their probabilities from above: (P1). Let v ∈ V (G). By the union bound: P r(G / ∈ (P1)) ≤ n · P r(d(v) ≥ 800 log n), and by Lemma 2.7 this gives: P r(G / ∈ (P1)) ≤ n enp 800 log n 800 log n ≤ n −799 = o(P r(G / ∈ (P0))).
(P2). The probability of |SM ALL(G)| ≥ n 0.3 is less than the probability that exists a set S of size n 0.3 with e(S, V \ S) ≤ d 0 · n 0.3 , so by the union bound: By Lemma 2.7 we can estimate the latter expression from above by: Finally, since 100e 0.001·0.9 0.001 ≤ e 0.015 , and since ω(n) = o(n 0.3 log n), we obtain: (P3). The probability that exist u, v ∈ V (G) such that u, v ∈ SM ALL(G), dist(u, v) = k is at most the probability that there is a path P of length k between them. By the union bound this is at most n k + 1 p k · P r(Both endpoints of P have degree at most d 0 | P ∈ G).
The probability that two endpoints u, v of a path both have at most d 0 neighbours is at most the probability that they have at most 2(d 0 − 2) edges that are not (u, v) or a part if the path. By Lemma 2.7 we have Apply the union bound once more, and sum over 1 ≤ k ≤ 4: (P6). By the union bound: log m 2 P r Bin n 2 log n , p < 1 2 n ≤ e o(n) P r Bin n 2 log n , log n n < 1 2 n .
(P7). By Lemma 2.3, a non-Hamiltonian n 4 , 2 -expander has at least n 2 32 boosters. By the union bound, the probability of the existence of such a subgraph Γ ⊆ G, with none of its boosters being an edge of G, is at most 2. Γ 0 has at most d 0 n edges, log n : e Γ 0 (U, W ) ≥ 1. Proof. Consider the following construction of a random subgraph Γ 0 of G with at most d 0 n edges and minimum degree at least min{δ(G), d 0 }: Clearly, Γ 0 has at most d 0 n edges and minimum degree at least min{δ(G), d 0 }, and every vertex not in SM ALL(G) has degree at least d 0 . We will show that with positive probability (and in fact, with high probability) a random subgraph Γ 0 constructed in this manner is such that for every pair of disjoint sets U, W ⊆ V (G), with |U | = |W | = n √ log n , the graph Γ 0 has an edge between U and W , thus ensuring the existence of the requested subgraph. By the union bound, the probability that exist such U, W with no edge between them is at most Proof. We will show that the subgraph Γ 0 constructed in Lemma 3.2 is an n 4 , 2 -expander. Since is has at most d 0 n edges, this will finish the proof. Let U ⊆ V (G), |U | = k ≤ n 4 . We will show that indeed |N Γ 0 (U )| ≥ 2k. Denote U 1 = U ∩ SM ALL(G), U 2 = U \ U 1 and let k 1 , k 2 be the sizes of U 1 , U 2 , respectively. Consider two cases: log n , since otherwise the sets U 2 , V (G) \ N (U 2 ) contradict the conclusion of Lemma 3.2. By (P2) we know that k 1 ≤ |SM ALL(G)| ≤ n 0.3 , so overall: 2. k 2 ≤ n √ log n . By Lemma 3.2 we know that δ(Γ 0 ) ≥ 2, and by (P3) we know that no two vertices in U 1 have distance less than 5. More specifically, every vertex of U 1 has at least two neighbours, and no two vertices share any common neighbour, so |N Γ 0 (U 1 )| ≥ 2k 1 . Next, we know that all vertices in U 2 have degree at least d 0 in Γ 0 , and by (P4) we have e Γ 0 (U 2 ) ≤ e G (U 2 ) ≤ k 2 · log 3 4 n, so: This means that |N Γ 0 (U 2 )| ≥ k 2 · log 1 4 n, since otherwise U 2 , N Γ 0 (U 2 ) contradict (P5). Finally, observe that, since the vertices of U 1 are at distance at least 5 from each other, we have: So overall: ✷ Corollary 3.1. Let G be a graph such that (P0)-(P7) hold. Then G is Hamiltonian.
Proof. By Lemma 3.3, G contains a subgraph Γ 0 which is an n 4 , 2 -expander with at most d 0 n edges. By (P7), if Γ 0 is not Hamiltonian then it has a booster in E(G) \ E(Γ 0 ). Add one such booster to Γ 0 to obtain a new subgraph of G, Γ 1 . This new subgraph is also an n 4 , 2expander, this time with at most d 0 n + 1 edges, and so is either Hamiltonian or has a booster in E(G) \ E(Γ 1 ). This process can be repeated until we find a subgraph Γ i which is Hamiltonian. Since in each step the length of a longest path grows by at least 1, at most n steps are required, so e(Γ i ) = e(Γ 0 ) + i ≤ d 0 n + n, which means the process can always continue until Hamiltonicity is achieved. ✷ Theorem 1 for p ≤ 100 log n n is obtained directly from Lemma 3.1 and Corollary 3.1.
(Q2). The probability that SM ALL(G) contains more than one vertex is at most the probability that exist u, v ∈ V (G) such that e({u, v}, V (G) \ {u, v}) ≤ 2t 0 . So by the union bound: which is sufficiently small.
Proof. Denote by v a vertex of G such that d(v) = δ(G), and by G ′ the graph obtained from G by removing v and all its edges. We observe that the properties (R2), (R3) hold for G ′ , since removing vertices does not affect these properties. Furthermore, by property (R1) we have δ(G ′ ) ≥ np 10 − 1. We claim that these are sufficient for showing that κ(G ′ ) ≥ np 30 . Indeed, suppose towards contradiction that there exists a set U ⊆ V (G ′ ) of size np 30 such that removing the vertices of U disconnects G ′ , and denote by W 1 , W 2 two components in the resulting graph (WLOG we assume |W 1 | ≤ |W 2 |). Consider the following cases: a contradiction.
Now, from (R3) we get that α(G ′ ) ≤ np 40 < κ(G ′ ). By Theorem 2.5 this means that G ′ is Hamiltonconnected. We now return to G. By (R0), v has at least two neighbours, say u 1 , u 2 . Since G ′ is Hamiltonconnected, it contains a Hamilton path P with u 1 , u 2 being its two endpoints. Now (u 1 , v), (v, u 2 ), P is a Hamilton cycle in G. ✷

Perfect matching
We provide a sketch of a proof for Theorem 2.
Observe that similarly to the case of Hamilton cycles, for the very sparse case, here defined as p(n) such that np − log n does not tend to infinity, the result is already known. This is due to the classical result by Erdős and Rényi, stating: Which means that lim n→∞ Pr(G contains no perfect matching) = lim n→∞ Pr(δ(G) = 0) = C > 0, which suffices. We also observe that the very dense case of p ≥ 0.01 was proven in Section 3.3 up to a small adjustment. Recall that in the proof of Lemma 3.8 we showed that in a graph with properties (R1)-(R3), removing a minimum degree vertex yields a Hamilton-connected graph. Replace the property (R0) with a new property (R0'): δ(G) > 0. By taking the minimum degree vertex and a Hamilton path starting at one of it's neighbours, we get a Hamilton path in G, which becomes a perfect matching by taking every other edge. Since like (R0), P r(G / ∈ (R0')) = (1 − p) (1+o(1))n , we obtain the desired result. Towards the goal of providing a proof idea for other ranges of p(n), we introduce the notion of staples, which will be used here similarly to the way boosters were used in the proof of Theorem 1: This enables us to take an approach similar to the proof in Section 3, following similar steps: (i) Show that for G ∼ G(n, p), the probability that G does not contain a relatively sparse (k, 1)expanding subgraph, for some k linear in n, is (1 + o(1))P r(δ(G) = 0); (ii) Show that the probability that G does not contain a staple for each of its sparse (k, 1)expanding subgraphs is o(P r(δ(G) = 0)).