Ore-degree threshold for the square of a Hamiltonian cycle

A classic theorem of Dirac from 1952 states that every graph with minimum degree at least n/2 contains a Hamiltonian cycle. In 1963, P\'osa conjectured that every graph with minimum degree at least 2n/3 contains the square of a Hamiltonian cycle. In 1960, Ore relaxed the degree condition in the Dirac's theorem by proving that every graph with $deg(u) + deg(v) \geq n$ for every $uv \notin E(G)$ contains a Hamiltonian cycle. Recently, Ch\^au proved an Ore-type version of P\'osa's conjecture for graphs on $n\geq n_0$ vertices using the regularity--blow-up method; consequently the $n_0$ is very large (involving a tower function). Here we present another proof that avoids the use of the regularity lemma. Aside from the fact that our proof holds for much smaller $n_0$, we believe that our method of proof will be of independent interest.


Notation and Definitions
Given a graph G, we denote the vertex set and edge set by V (G) and E(G) respectively, when the graph G is clear by the context we refer to them as V and E respectively. When uv ∈ E(G) we denote it by u ∼ v otherwise u v. We denote a cycle on t vertices by C t and a path on t vertices by P t . When G is a graph on n vertices and C n ⊆ G, we call C n a Hamiltonian cycle. A bipartite graph G = (V, E), where V = A ∪ B, A ∩ B = ∅ will be denoted by G (A, B). The balanced complete r-partite graph with color classes of size t is denoted by K r (t).  is the density of the graph between A and B. We write d(A) = 2e(A)/|A| 2 . A graph G on n vertices is γ-dense if it has at least γ n 2 edges. A bipartite graph G(A, B) is γ-dense if it contains at least γ|A||B| edges. Throughout the paper log denotes the base 2 logarithm.

Powers of Cycles
A classical result of Dirac [8] asserts that if G is a graph on n ≥ 3 vertices with δ(G) ≥ n/2, then G contains a Hamiltonian cycle. Note that when n = 2t, Dirac's theorem implies that G contains t vertex disjoint copies of K 2 . In 1963, Corrádi and Hajnal [7] proved that if G is a graph on n = 3t vertices with δ(G) ≥ 2n 3 , then G contains t vertex disjoint triangles. Generalizing the Corrádi-Hajnal theorem, Erdős conjectured [9] and Hajnal and Szemerédi later proved [17] the following: Theorem 1.1 (Hajnal-Szemerédi). Let G be a graph on n = t(k + 1) vertices. If δ(G) ≥ kn k+1 , then G contains t vertex disjoint copies of K k+1 .
Finally in 1976, Bollobas and Eldridge [2], and indpendently Catlin [4], made a conjecture which would generalize the Hajnal-Szemerédi theorem: If G and H are graphs on n vertices with ∆(H) ≤ k and δ(G) ≥ kn−1 k+1 , then H ⊆ G. While this conjecture is still open in general, we will only be interested in the k = 2 case which was proved by Aigner and Brandt in 1993 [1]  Note that all of these degree conditions are easily seen to be best possible. Let H be a graph with vertex set V . The k th power of H, denoted H k , is defined as follows: V (H k ) = V and uv ∈ E(H k ) if and only if the distance between u and v in H is at most k. When k = 2 we call H 2 the square of H. For notational convenience we call the k th power of a cycle a k-cycle (k-path is analogous). Notice that C k−1 n contains n k vertex disjoint copies of K k . Furthermore, notice that C 2 n contains every graph H on n vertices with ∆(H) ≤ 2 (actually P 2 n also has this property). In 1963, Pósa made a conjecture (see [9]) that would significantly strengthen the Corrádi-Hajnal theorem (and retroactively Theorem 1.2, see [13]).
After Erdős' conjecture became the Hajnal-Szemerédi theorem, Seymour made a conjecture in 1974 [30] which generalizes Pósa's conjecture to handle all values of k (note that for k ≥ 4, this does not generalize the Bollobás-Eldridge, Catlin conjecture). Conjecture 1.4 (Seymour). Let G be a graph on n vertices. If δ(G) ≥ kn k+1 , then C k n ⊆ G.
Starting in the 90's a substantial amount of progress was made on these conjectures. Jacobson (unpublished) first established that the square of a Hamiltonian cycle can be found in any graph G given that δ(G) ≥ 5n/6. Later Faudree, Gould, Jacobson and Schelp [16] improved the result, showing that the square of a Hamiltonian cycle can be found if δ(G) ≥ (3/4 + ε)n. The same authors further relaxed the degree condition to δ(G) ≥ 3n/4. Fan and Häggkvist lowered the bound first in [10] to δ(G) ≥ 5n/7 and then in [11] to δ(G) ≥ (17n + 9)/24. Faudree, Gould and Jacobson [15] further lowered the minimum degree condition to δ(G) ≥ 7n/10. Then Fan and Kierstead [12] achieved the almost optimal δ(G) ≥ 2 3 + ε n. They also proved in [13] that already δ(G) ≥ (2n − 1)/3 is sufficient for the existence of the square of a Hamiltonian path. Finally, they proved in [14] that if δ(G) ≥ 2n/3 and G contains the square of a cycle with length greater than 2n/3, then G contains square of a Hamiltonian cycle.
For Conjecture 1.4, in the above mentioned paper of Faudree et al in [16], it is proved that for any ε > 0 and positive integer k there is a C such that if graph G, on n vertices, satisfies δ(G) ≥ 2k−1 2k + ε n, then G contains the k th power of a Hamiltonian cycle. Using the regularity-blow-up method first in [23] Komlós, Sárközy and Szemerédi proved Conjecture 1.4 in asymptotic form, then in [21] and [24] they proved both conjectures for n ≥ n 0 . The proofs used the regularity lemma [31], the blow-up lemma [22,25], and the Hajnal-Szemerédi theorem [17]. Since the proofs used the regularity lemma the resulting n 0 is very large (it involves a tower function). A new proof of Pósa's conjecture was given by Levitt, Sárközy and Szemerédi [27] which avoided the use of the regularity lemma and thus significantly decreased the value of n 0 . An explicit bound on n 0 was determined by Châu, DeBiasio, and Kierstead in [6]; however, for small n 0 the conjecture is still open. Finally, Jamshed and Szemerédi [18] gave a new proof of the Seymour's conjecture that avoided the use of the regularity lemma.

Ore-type generalizations of Dirac-type results
For a pair of non-adjacent vertices (u, v), the value of deg(u) + deg(v) is called the Ore-degree of (u, v). We denote by δ 2 (G) the minimum Ore-degree over all non-adjacent pairs of vertices in G. In 1960, Ore proved that if G is graph on n ≥ 3 vertices with δ 2 (G) ≥ n, then G contains a Hamiltonian cycle. Since any graph with δ(G) ≥ n 2 satisfies δ 2 (G) ≥ n, Ore's theorem strengthens Dirac's theorem. Inspired by this, researchers have sought to generalize minimum degree ("Diractype") conditions to Ore-type degree conditions; for a survey of such results see [20].
Two important examples of Ore-type results are the following generalizations of Theorem 1.1 and 1.2. Theorem 1.5 (Kierstead-Kostochka [19]). Let G be a graph on n = t(k + 1) vertices. If δ 2 (G) ≥ 2kn k+1 − 1, then G contains t vertex disjoint copies of K k+1 . Theorem 1.6 (Kostochka-Yu [26]). Let G and H be graphs on n vertices. If ∆(H) ≤ 2 and A natural Ore-type generalization of Pósa's conjecture suggests that if δ 2 (G) ≥ 4n 3 , then G contains a Hamiltonian 2-cycle. It turns out that this natural generalization is not quite true as Châu [5] gave a construction of a graph G for which δ 2 (G) = 4n 3 , but G does not contain the square of a Hamiltonian cycle. However, in the same paper, Châu uses the regularity-blow-up method to prove that if G is a graph on n ≥ n 0 vertices with δ 2 (G) > 4n 3 , then C 2 n ⊆ G. In fact, he is able to give an even more refined degree condition: Theorem 1.7 (Châu). Let G be a graph on n vertices. If δ 2 (G) ≥ 4n−1 3 and (i) δ(G) ≤ n 3 + 2, then P 2 n ⊆ G.
(See [5], Proposition 9.1 for an explanation of why this result actually implies Theorem 1.6 and the k = 2 case of Theorem 1.5 for sufficiently large n despite the fact that 4n−1 . . . One of the purposes of this paper is to present another proof of Theorem 1.7.(ii) which avoids the use of the regularity lemma, thus resulting in a much smaller value of n 0 . Theorem 1.8. There exists n 0 such that if G is a graph on n ≥ n 0 vertices with then C 2 n ⊆ G.
Aside from lowering the bound on n 0 , we believe that the techniques used in this paper are of independent interest and can have more applications. In particular, our proof provides a simpler template for approaching the following Ore-type version of Conjecture 1.4. Conjecture 1.9 (Châu). Let G be a graph on n vertices. If δ 2 (G) ≥ 2kn−1 k+1 and δ(G) > (k−1)n k+1 + 2, then C k n ⊆ G.

Outline of the Proof
As is common in these types of problems, our proof is divided into extremal and non-extremal cases. The extremal conditions will resemble the properties found in Figure 1; either there is a vertex close to smallest possible degree, or there is a set of size approximately n/3 with very few edges. We formally define the extremal conditions below. Definition 1.10 (Extremal Condition 1). We say that G satisfies extremal condition 1 with parameter α if there exists v ∈ V (G) such that deg(v) < ( 1 3 + α)n.
Definition 1.11 (Extremal Condition 2). Let 0 < α 1 3 . The graph G satisfies extremal condition 2 with parameter α if there exists disjoint sets A 1 , A 2 such that for i = 1, 2, |A i | ≥ (1/3 − α)n and d(A i ) < α. Definition 1.12 (Extremal Condition 3). Let 0 < α 1 3 . The graph G satisfies extremal condition 3 with parameter α if there exists a set A 1 such that |A 1 | ≥ (1/3 − α)n, d(A 1 ) < α, and for all . If G does not satisfy extremal condition 1,2, or 3 with parameter α, then we say G is not α-extremal. Specifically, this means that δ(G) ≥ (1/3 + α)n and for all These extremal cases are dealt with in [5] without the use of the regularity lemma; however, the blow-up lemma is used in multiple cases. Each time the blow-up lemma is used, a more elementary argument could have sufficed. In Section 2 we provide an alternate argument which can be used in [5] instead of the blow-up lemma.
The non-extremal case is where our proof differs most significantly from [5] and is the main focus of our paper. We avoid the use of the regularity lemma, the blow-up lemma, and Theorem 1.5 by instead using Erdős-Stone type results to cover all but a small fraction of the vertex set with disjoint balanced complete tripartite graphs of size about log n. Then we prove a new connecting lemma which allows us to connect the complete tripartite graphs by square paths. Aside from any leftover vertices, we have a nearly spanning structure which contains a square cycle and is quite robust in the sense that most of the vertices are in complete tripartite graphs on size log n. Finally, we take advantage of the robustness of our structure by inserting the leftover vertices in such a way that the resulting structure contains a spanning square cycle. All of this will be made precise in Section 3.

Extremal case
In [5], the extremal cases are handled with very detailed, yet elementary arguments -with one exception. In many of the cases and subcases in [5] the problem of finding a Hamiltonian square cycle is reduced to finding a Hamiltonian square cycle in a balanced tripartite graph where each pair is nearly complete, with the exception of a small number of vertices which still satisfy some minimum degree condition. Here Châu uses the fact that these very dense pairs are (ε, δ)-super regular so the blow-up lemma can be applied to show that the desired square cycle exists. However, the property that these dense pairs have is far stronger than the property of being (ε, δ)-super regular, so the application of the blow-up lemma is unnecessary.
Our goal in this section is simply to provide an elementary argument which could be used to replace all of the uses of the blow-up lemma in the extremal cases of [5]. Note that we will not reproduce the proof found in [5], as we are only providing a minor diversion to the conclusion of certain cases of the argument.
then we can cover V (H) by disjoint triangles such that each triangle uses exactly one vertex in each A i .
Proof. We first find a perfect matching M 1 between A 1 and A 2 by an application of the König-Hall theorem. Then we find a perfect matching between M 1 and A 3 , such that e = xy ∈ M 1 is matched with a vertex z ∈ N (x, y, A 3 ). For any edge e = xy ∈ M 1 we have deg(x, y, A 3 ) ≥ (1 − 2α )m, therefore, by König-Hall theorem there exists a perfect matching between M 1 and A 3 as desired. Proof. Let t = (x 1 , x 2 , x 3 ) and t = (y 1 , y 2 , y 3 ) be any two triangles in T such that is a square-path). We say that {t, t } is a good pair, if t precedes t and t precedes t. By the degree conditions above, any t i ∈ T makes a good pair with at least (1 − √ α )m other triangles in T . Make an auxiliary graph H over T such that each triangle t i ∈ T is adjacent to the triangle t j if and only if {t i , t j } is a good pair. By the above observation we clearly have δ(H ) > m/2, hence by the Dirac's theorem there is a Hamiltonian cycle in H . Also since δ(H ) > m/2, H is Hamiltonian connected and thus there is a Hamiltonian path in H which starts with t 1 and ends with t m . It is easy to see that this Hamiltonian cycle (path) in H corresponds to the square of a Hamiltonian cycle (path) in H.
Finally we arrive at the main lemma which can be used to replace the use of the blow-up lemma in the extremal cases of [5]. Lemma 2.3. Let 0 < α β γ 1 and let H be a balanced tripartite graph on 3m = n ≥ n 0 vertices with V (H) partitioned as A 1 , A 2 , A 3 . If for all i = j, there are at least (1 − β)m vertices in A i with at least (1 − α )m neighbors in A j and δ(A i , A j ) ≥ γm, then H contains the square of a Hamiltonian cycle. Furthermore, if we specify two edges u 1 u 2 and u 3m−1 u 3m such that for all Proof. Call a vertex u in A i bad if u has less than (1 − α )m neighbors in A j for some j = i. By the hypothesis, there are at most 2βm bad vertices in each A i . Now with a simple greedy procedure, for each bad vertex u ∈ A 1 we find a triangle t 2 = (b 1 , b 2 , b 3 ), such that b 1 = u and b 2 and b 3 are typical (not bad) vertices in A 2 and A 3 . We find two more similar triangles t 1 = (a 1 , a 2 , a 3 ) and We replace these three triangles with an exceptional triangle (d 1 , d 2 , d 3 ) with one vertex each in A 1 , A 2 and A 3 , such that for 1 ≤ i ≤ 3, d i is connected to common neighbors of a i and c i . By the fact that a i and c i are not bad vertices every d i has at least (1 − 3α )m neighbors in both of the other two sets. We similarly make an exceptional triangle for the remaining bad vertices. Since the total number of bad vertices is at most 6βm and the minimum degree is γm 6βm, this greedy procedure can be easily carried out. In the remaining parts of A 1 , A 2 , and A 3 by Lemma 2.1 we find a triangle cover and add all the exceptional triangles to the cover. Then by Lemma 2.2, we find the square of a Hamiltonian cycle. Now suppose u 1 u 2 and u 3m−1 u 3m are given edges such that for all . Now by applying Lemma 2.1 we find a triangle cover and add all the exceptional triangles to the cover. Then by Lemma 2.2, we find the square of a Hamiltonian path which starts with t 1 and ends with t m .

Non-extremal case
Before we give an overview of the non-extremal case, it would be helpful to have some idea of how the non-extremal case is proved in [5] (which is a generalization of the arguments in [21], [23], [24]). Suppose G is a non-extremal graph on n vertices (n sufficiently large) with δ 2 (G) ≥ 4n 3 . Using the regularity lemma and Theorem 1.5, one can show that G contains a set of disjoint balanced 4-partite and 3-partite graphs spanning almost all of G each having size Ω(n). Each of these multi-partite graphs H has the property that every pair of color classes forms a suitably dense psuedorandom bipartite graph, so by applying the blow-up lemma, one obtains an almost spanning square path in H. If we connect these multi-partite graphs together with square paths before applying the blow-up lemma, one will obtain an almost spanning square path of G. Finally the remaining vertices need to somehow be inserted, which is an elementary, but detailed argument.
We are able to avoid the regularity-blow-up method by showing that for sufficiently large n (but nowhere near as big as needed for the regularity lemma), G can be partitioned into disjoint balanced complete tripartite graphs spanning almost all of G, each having size Ω(log n); we call this "the cover" and it is built in Section 3.1. Since the tripartite graphs are complete, we do not have to apply the blow-up lemma; if we go around a complete tripartite graph picking vertices from each of the color classes sequentially we get a square-path. Next we must prove a Connecting Lemma which allows us to connect the tripartite graphs by short square-paths giving us a "cycle of cliques"; this is done in Section 3.2. At the end of this process there will be a few leftover vertices which need to be inserted; this is done in Section 3.3.
Here is the statement of the non-extremal case (notice that in the non-extremal case we are able to slightly relax the Ore-degree condition).
and G is not α-extremal, then C 2 n ⊆ G.

The Cover
In order to cover most of the vertices in G with complete tripartite graphs as mentioned above, we will need quantitative versions of some classical results in extremal graph theory.

Lemmas
Fact 3.2. Let 0 < d, γ < 1. If G(A, B) is a (d + 2γ)-dense bipartite graph, then there must be at least γ|B| vertices in B for which the degree in A is at least (d + γ)|A|.
Proof. Indeed, otherwise the total number of edges would be less than Lemma 3.3. Let 0 < c, γ < 1/3, s = c log n , and let G be a graph on n ≥ n 0 vertices with By the degree condition, each vertex in B 1 has at least γs neighbors in each A i . There are at most 2 |A| = 2 3s = n 3c different possible neighborhoods, so by averaging there must be a neighborhood that appears for a set B 2 of at least |B 1 | n 3c ≥ γ 2 n n 3c = γ 2 n (1−3c) vertices of B 1 . Selecting an appropriate subset B of B 2 , we get the desired complete K 4 (γs).
We need a version of the Erdős-Stone theorem where we have control of the parameters. While there are a sequence of improvements by Bollobás-Erdős, Bollobás-Erdős-Simonovits, and Bollobás-Kohayakawa (to name a few), we will state a version due to Nikiforov [28] which gives an explicit lower bound on n. Finally, we need a simple fact which allows us to translate our Ore-degree condition into an appropriate edge density condition. Proof. Define γ so that e(G) = γ n 2 and suppose δ 2 (G) ≥ 2d(n − 1). We have 3.1.2 Building the cover Definition 3.6 (Tripartite Cover). Let s, n ∈ Z + . A (s, n ) tripartite cover is a collection T of vertex disjoint copies of Note that in the following lemma we do not assume that G is non-extremal.
Proof of Lemma 3.7. Set t 0 = η 6 64 log n and c 0 = η 2 . By (2) and Fact 3.5 we have e(G) ≥ ( 2 3 −ε) n 2 2 . We repeatedly apply Lemma 3.4 with c = η 2 to find complete tripartite graphs with each color class of size t 0 until the remaining graph contains no copy of K 3 (t 0 ). Let T be the collection of tripartite graphs obtained in this way, and let U = V (G) \ V (T ), where V (T ) = T ∈T V (T ). If |U | < ηn, then we are done, so suppose |U | ≥ ηn.
does not contain a copy of K 3 (t i ), in which case there exists a cover T i+1 such that |V (T i+1 )| ≥ |V (T i )| + η 4 n and every color class in the cover has size between t i and 2t i .
It is clear that if Claim 3.8 holds, then by applying the claim j times for some j ≤ 1 η 4 , T j will satisfy the conclusion of Lemma 3.7. We now finish the proof of the cover lemma by proving Claim 3.8.
Proof. Let 0 ≤ i ≤ 1 η 4 and suppose G[U i ] does not contain a copy of K 3 (t i ). In this case by Lemma Start by setting Z = ∅. We will consider each T ∈ T i one by one. If d(U i , T ) < ( 2 3 + 6η 2 ), then consider the next element of T i . If d(U i , T ) ≥ ( 2 3 + 6η 2 ), then by Lemma 3.3 there exists , which can be split into four copies of K 3 (η 2 t i ) = K 3 (t i+1 ). Move the used vertices from U i into Z and reset U i := U i \ Z. Let T i be the set of 3-partite graphs in T i for which the procedure succeeded. If |T i | ≥ η 2 n 3t i , then we will have increased the cover by at least 3η 2 t i · η 2 n 3t i = η 4 n. If |U i | < ηn or we have increased the cover by η 4 n, we partition each color class into parts of size at least t i+1 (which implies that all parts have size at most 2t i+1 ).
So suppose we have increased the cover by less than η 4 n and we still have |U i | ≥ ηn. In this case we have |T i | < η 2 n 3t i which implies For every T ∈ T i \ T i , we have Now by (4) and (5) we have By (6) and (7), we have ( 2

Connecting
In this section, we will make use of the non-extremality of G.

Connecting triangles
Definition 3.9. Given disjoint triangles T and T in a graph G, we say T is square-connected to T if there exists x 1 y 1 ∈ E(T ), x 2 y 2 ∈ E(T ), and a square path Q ⊆ G − (V (T ) ∪ V (T )) such that x 1 y 1 Qx 2 y 2 is a square path. Furthermore, we say T is square-adjacent to T if there exists x 1 y 1 ∈ E(T ), x 2 y 2 ∈ E(T ) such that x 1 y 1 x 2 y 2 is a square path.
We start by proving the following simple, but useful proposition.  (ii) If T is not square-adjacent to T , then there are vertices x 1 , y 1 ∈ T and x 2 , y 2 ∈ T such that x 1 x 2 and y 1 y 2 .
Proof. (i) Since e(T, T ) ≥ 5, there exists a vertex y 1 ∈ T with deg(y 1 , T ) ≥ 2 and a vertex (ii) If T is not square-adjacent to T , then by part (i) we have e(T, T ) ≤ 4. If e(T, T ) ≤ 3 it is easy to verify that the statement holds, so suppose e(T, T ) = 4. Now there exists a vertex z 1 ∈ T with deg(z 1 , T ) ≥ 2 and a vertex z 2 ∈ T with deg(z 2 , T ) ≥ 2. If z 1 ∼ z 2 , this would imply that T is square-adjacent to T as in part (i). So we have z 1 z 2 and thus deg(z 1 , T ) = 2 = deg(z 2 , T ). Since e(T, T ) = 4, we may pair up the remaining vertices x 1 , y 1 and x 2 , y 2 such that x 1 x 2 and y 1 y 2 .
Lemma 3.11 (Connecting Lemma). For all 0 < ε α 1 there exists n 0 such that if G is a graph on n ≥ n 0 vertices with δ 2 (G) ≥ ( 4 3 − 4ε)n such that G is not α-extremal, then the following statements hold:  (i) Let T and T be disjoint triangles in G and let G = G − T − T . If T is square-adjacent to T , then we are done, so suppose not. By Proposition 3.10, there are at least two disjoint non-adjacent pairs of vertices in T × T . Let (x i , y i ) and (x j , y j ) be two such pairs and define Consider two disjoint non-edges (x i , y i ), (x j , y j ) such that |C i,j | is maximum. We may label the vertices of T as x 1 , x 2 , x 3 and the vertices of T as y 1 , y 2 , y 3 such that the disjoint non-edges which maximize |C i,j | are (x 1 , y 1 ) and (x 2 , y 2 ); i.e.,

Together this gives
Now suppose for a contradiction that T is not square connected to T . Under this assumption, we have the following facts. Proof. (i) Suppose there is an edge c 1 c 2 ∈ G with c 1 , c 2 ∈ C. Then x 1 x 2 c 1 c 2 y 1 y 2 is a square path which connects T to T .
Suppose there is an edge bc ∈ G with b ∈ B and c ∈ C. Let i ∈ {1, 2} such that b ∼ x i , then x 3−i x i cby 1 y 2 is a square path which connects T to T .
Suppose there is an edge ac ∈ G with a ∈ A and c ∈ C. Let i ∈ {1, 2} such that a ∼ y i , then x 1 x 2 acy i y 3−i is a square path which connects T to T .
Suppose there is an edge ab ∈ G with a ∈ A and b ∈ B. Let i ∈ {1, 2} such that a ∼ y i and let j ∈ {1, 2} such that b ∼ x j . Then x 3−j x j aby i y 3−i is a square path which connects T to T .
By Claim 3.12(ii) and (8), we have Now we are ready to prove Lemma 3.11 (i). We consider two cases based on the density of G  |N (a 1 , b 1 , a 2 Since we are not in the extremal case, we have an edge is empty, so without loss of generality suppose G[B] is empty. By Claim 3.12(ii), |B| ≤ 1 and since |C| ≤ 1, (8) implies Recall that we are trying to connect T = x 1 x 2 x 3 to T = y 1 y 2 y 3 and we have not made use of x 3 or y 3 thus far. By (11) and δ(G) ≥ (1/3 + α)n, we have By Claim 3.12.(ii), |B|, |C| ≤ 1. Thus by (8), we have |A| ≥ ( 2 3 −9ε)n. Since deg(y i ) ≥ ( 2 3 −2ε)n for i = 1, 2, we have |N (y 1 , y 2 )| ≥ ( 1 3 − 4ε)n. If |N (y 1 , y 2 ) ∩ L| ≥ 3, then there is a triangle T in the common neighborhood of y 1 and y 2 . We may apply Lemma 3.11.(ii) to connect e to T with a square path on at most 12 vertices; thus the total length from e to e is at most 14 vertices. So suppose |N (y 1 , y 2 ) ∩ L| ≤ 2 and let z ∈ N (y 1 , y 2 Since we are not in the extremal case, there exists a 1 a 2 ∈ E(G[N (z) ∩ A]). Since a 1 ∈ A, there exists i ∈ {1, 2} such that a 1 ∼ y i . Thus y 3−i y i za 1 a 2 x 1 x 2 is the desired square path.
Finally, to obtain the stronger conclusion we first note that since deg( Depending on whether |N (x 1 , x 2 ) ∩ L| ≥ 3, |N (y 1 , y 2 ) ∩ L| ≥ 3, we will connect an edge or a triangle from N (x 1 , x 2 ) to an edge or triangle from N (y 1 , y 2 ) using one of the statements proved above. This will give us a square path from e to e having at most 16 vertices which can start with either direction of e and end with either direction of e .

Connecting the complete tripartite graphs
Given a tripartite cover {K 1 , . . . , K m }, we need to find "short" square paths connecting a triangle of K i to a triangle of K i+1 . If we simply use Lemma 3.11.(i) to connect a triangle of K 1 to a triangle of K 2 , then this will fix a direction for K 2 . So now we need to connect a directed edge of K 2 to a triangle in K 3 . Furthermore, when we connect K m to K 1 , both directions will be fixed. We now show how to apply Lemma 3.11 to achieve this goal.
Definition 3.13. Let K = (V 1 , V 2 , V 3 ) and K = (V 1 , V 2 , V 3 ) be two disjoint balanced complete tripartite graphs. We say that K is square-connected to K if there exists a triangle T ∈ K that is square-connected to a triangle T ∈ K , i.e. there exists a square path P = x 1 y 1 Qx 2 y 2 such that x 1 y 1 ∈ E(T ) and x 2 y 2 ∈ E(T ) and V (Q) ∩ (V (T ) ∪ V (T )) = ∅. When K is square-connected to K , we say that the square path P respects the orientation of K and Lemma 3.14. For all 0 < ε, c α 1 there exists n 0 such that if G is a graph on n ≥ n 0 vertices with δ 2 (G) ≥ ( 4 3 − 4ε)n and G is not α-extremal, then the following statement holds. Given disjoint balanced complete tripartite subgraphs K = (V 1 , V 2 , V 3 ) and K = (V 1 , V 2 , V 3 ) in G with color classes of size at least c log n, K is square-connected to K with a square path P on at most 16 vertices which respects the orientation of K and K .
is a clique. Case 1 There exists i, j ∈ {1, 2, 3} such that |V i ∩ L|, |V j ∩ L| ≥ 3. In this case we apply Lemma 3.11.(i) to connect a triangle T ∈ V i ∩ L to a triangle T ∈ V j ∩ L with a square path P on at most 10 vertices. Now if i = 1, then we take P = v 2 v 3 P , where v 2 and v 3 are arbitrary vertices in V 2 and V 3 . If i = 2, then we take P = v 2 v 3 P , where v 2 ∈ V (T ) is the vertex not in P and v 3 is an arbitrary vertex in V 3 . Finally, if i = 3, then we take P = v 2 P , where v 2 is an arbitrary vertex in V 2 . We similarly append P with one or two vertices from V 1 and V 2 . Note that |P | ≤ 14 and it respects the orientation of K and K .
Case 2 Not Case 1. Without loss of generality suppose that |V i ∩ L| ≤ 2 for all i ∈ {1, 2, 3}. Let v 2 ∈ V 2 \ L and v 3 ∈ V 3 \ L. First suppose that there exists j ∈ {1, 2, 3} such that |V j ∩ L| ≥ 3. In this case we apply Lemma 3.11.(ii) to connect v 2 v 3 to a triangle in V j ∩ L with a square path on at most 14 vertices. Similarly as above we append the path with one or two vertices from V 1 and V 2 to get the desired path.
Definition 3.15 (Connected tripartite cover). Let q, s, n ∈ Z + . A (q, s, n ) connected tripartite cover is a (s, n ) tripartite cover {K 1 , . . . , K m } together with a collection of m square paths {P 1 , . . . , P m } such that K i is square connected to K i+1 by P i where P i respects the orientation of K i and K i+1 and |V (P i )| ≤ q for all i ∈ [m]. Note that a (q, s, n ) connected tripartite cover contains a square cycle on n vertices. Lemma 3.16 (Connected cover lemma). For all 0 < ε, c η α 1 there exists n 0 such that if G is a graph on n ≥ n 0 vertices with δ 2 (G) ≥ ( 4 3 − 2ε)n such that G is not α-extremal, then G contains a (18, c log n, (1 − 2η)n) connected triangle cover.
Proof. First apply Lemma 3.7 to get a ( c ε log n, (1 − η)n) tripartite cover T = {K 1 , . . . , K m }. Fix an orientation for each tripartite graph in T and applying Lemma 3.14 connect and append it with v 3 ∈ V 3 so as we use at least one triangle each from K i and K i+1 . We fix these two triangles and make their vertices forbidden to be used for any further connection. Similarly all other vertices of P i are forbidden. Furthermore, if some K ∈ T has more than ε · c ε log n forbidden vertices we make all the vertices in K forbidden. Note that by the end the number of forbidden vertices are at most 18 · n c log n < εn, therefore at any time in the remaining graph we still have deg(u) + deg(v) ≥ ( 4 3 − 4ε)n, hence we can continue to apply Lemma 3.14. Remove all vertices from the tripartite graphs that are part of some P i except the starting and ending triangles and rebalance the tripartite graphs by discarding arbitrary subset of vertices from larger color classes to get the desired (18, c log n, (1 − 2η)n) connected tripartite cover.

Inserting the remaining vertices
Finally we show that if we are given a connected tripartite cover, we can assign the remaining vertices to the tripartite graphs in such a way that they can be incorporated into a square cycle. Proof. Let U = V (G) − V (K) − V (P) and note that |U | ≤ 2ηn. We will try to assign the vertices of U to the complete tripartite graphs, but in the process we will end up having to modify the original cover. For convenience, we let the original cover consist of complete tripartite graphs {T 1 , . . . , T m } and square paths {P 1 , . . . , P m } where 1 6c n log n ≤ m ≤ 1 3c n log n and throughout the process, we will refer to the tripartite graphs by these same names even if they are modified. We assume that size of a color class in T i is t. However, we will maintain a set T * of triangles which cannot be modified as they are being used to insert vertices into some T i .
Let w ∈ U . We will prove that we can assign w to some T i while only adding at most 8 triangles to T * . Once η 1/3 c log n vertices have been assigned to T i , then we make all of the vertices of T i forbidden. Since there are 2ηn vertices to be assigned, this will make at most 2ηn η 1/3 c log n ≤ 12η 2/3 m tripartite graphs forbidden and a total of at most 12η 2/3 m · 6c log n ≤ 24η 2/3 n forbidden vertices Z. For any vertex we only consider its neighborhood in V (G) − V (T * ) − V (P) − Z so for the rest of the proof we will assume that First, if w has at least 2 neighbors in every color class of T i = (T i 1 , T i 2 , T i 3 ), then there are two triangles (x 1 , x 2 , x 3 ) and (y 1 , y 2 , y 3 ) in N (w), such that x j , y j ∈ T i j . Clearly we can assign w to T i . We add the two triangles (x 1 , x 2 , x 3 ) and (y 1 , y 2 , y 3 ) to T * ; we say that these triangles are blocked by w.
By the degree condition, deg(w ) ≥ ( 2 3 − √ η)n. So we may insert w into T i adding two triangles to T * and try to insert w instead. So we assume deg(w) ≥ ( 2 3 − √ η)n and we will try to insert w by adding at most six triangles to T * . We may also assume that for all v ∈ {w} ∪ R(w), deg(v) < ( 2 3 + √ η)n, otherwise v will have at least two neighbors in each color class of some T i , in which case we can move v to T i and replace v with w. This implies that for all v ∈ V (G), if there exists u ∈ {w} ∪ R(w) such that v ∼ u, then Since deg(w) ≥ ( √ η)t} and R (w) = i∈I T i 3 \ N (w). Note that |R (w)| ≥ (1 − 2 4 √ η)tm ≥ ( 1 3 − α)n and since G is not α-extremal, e(R (w)) ≥ αn 2 . At least at least two neighbors in every color class of some T ∈ T , then we could move v to T , replace it with a vertex from X i (which has at least α t neighbors in T j 2 and T j 3 ) and replace the vertex from X i with w; thus e(T j 1 , T ) ≤ 2t 2 for all T = T i .

Proof of Theorem 3.1
Given G we first apply Lemma 3.16 to get a (18, c log n, (1 − 2η)n) connected cover in G. We insert the remaining vertices into the cover using Lemma 3.17 and get a set T of m disjoint complete tripartite graphs, a set of square paths P, and the function f : V (G) − V (T ) − V (P) → T . Note that for any w such that f (w) = T i , there are two triangles blocked by w. Let (x 1 , x 2 , x 3 ) and (y 1 , y 2 , y 3 ) be the triangles blocked by w, notice that by construction x i , y i ∈ N (w). Create an auxiliary triangle (z 1 , z 2 , z 3 ) in T i to replace these two triangles and connect z i to the common neighbors of x i and y i . Note that the modified T i is still a complete tripartite graph. We similarly introduce such an auxiliary triangle for each vertex w ∈ V (G) − V (T ) − V (P). Find a triangle cover in the remaining part of T i except for the two triangles that are part of P i−1 and P i by an application of Lemma 2.1. Combining these triangles with the auxiliary triangles, we find a Hamiltonian square path by applying Lemma 2.2 that starts with the last triangle t i−1 of P i−1 and ends with the first triangle t i of P i .