Matchings of quadratic size extend to long cycles in hypercubes

Ruskey and Savage in 1993 asked whether every matching in a hypercube can be extended to a Hamiltonian cycle. A positive answer is known for perfect matchings, but the general case has been resolved only for matchings of linear size. In this paper we show that there is a quadratic function $q(n)$ such that every matching in the $n$-dimensional hypercube of size at most $q(n)$ may be extended to a cycle which covers at least $\frac34$ of the vertices.


Introduction
Frank Ruskey and Carla Savage in 1993 formulated the following problem [11]: Does every matching in a hypercube extend to a Hamiltonian cycle? More than two decades have passed and the question still remains open. It may be of interest that a complementary problem on the existence of a Hamiltonian cycle avoiding a given matching in a hypercube has been already resolved [4].
An important step towards a solution to the Ruskey-Savage problem was made in 2007 by Fink who answered the question affirmatively for every perfect matching [7]. Note that this implies a positive solution for every matching that extends to a perfect one, which includes e.g. every induced matching [12]. However, this result does not immediately provide a complete answer to the original question, as hypercubes contain matchings that are maximal with respect to inclusion, but still not perfect. Actually, to determine the minimum size of such a matching is another long-standing problem [9]. The simplicity and elegance of Fink's method inspired several generalizations [1,6,10], but none of them addresses the extendability of imperfect matchings.
As far as arbitrary matchings are concerned, there are only partial results that deal with matchings of linear size. The author of the current paper showed that a set P of at most 2n − 3 edges of the ndimensional hypercube extends to a Hamiltonian cycle iff P induces a linear forest [5]. This bound is sharp, as for every n ≥ 3 there is a non-extendable linear forest of 2n − 2 edges [3]. Of course, these edges do not form a matching, so this does not imply a negative answer to the Ruskey-Savage problem. In the case when P is a matching, the bound on |P| was improved to 3n − 10 by Wang and Zhang [14]. * Financial support from the Czech Science Foundation under the grant GA14-10799S is gracefully acknowledged. The purpose of this paper is to derive a quadratic upper bound on the size of a matching that extends to a cycle which covers at least 3 4 of the vertices of the hypercube. Our result is based on an inductive construction which combines a refinement of Fink's method for perfect matchings [8] with a lemma on hypercube partitioning due to Wiener [15].

Preliminaries
The graph-theoretic terms used in this paper but not defined below may be found e.g. in [2]. Throughout the paper, n always denotes a positive integer while [n] stands for the set {1, 2, . . . , n}.
Vertex and edge sets of a graph G are denoted by V (G) and E(G), respectively. A sequence a = x 1 , x 2 , . . . , x n+1 = b of pairwise distinct vertices such that x i and x i+1 are adjacent for all i ∈ [n] is a path between a and b of length n. We denote such a path and its vertices by P ab and V (P ab ), respectively. Let P ab and P bc be paths such that V (P ab ) ∩ V (P bc ) = {b}. Then P ab + P bc denotes the path between a and c, obtained as a concatenation of P ab with P bc (where b is taken only once). Observe that the operation + is associative. A cycle of length n is a sequence x 1 , x 2 . . . , x n of pairwise distinct vertices such that x 1 is adjacent to x n and x i is adjacent to x i+1 for all i ∈ [n]. The sets of vertices x 1 , x 2 . . . , x n and edges x 1 x 2 , x 2 x 3 , . . . , x n x 1 of a cycle C are denoted by V (C) and E(C), respectively.
In this paper we deal with the n-dimensional hypercube Q n which is a graph with all n-bit strings as vertices, an edge joining two vertices whenever they differ in a single bit. Given a string • if Q D (u) and Q D (v) are adjacent subcubes, then an arbitrary vertex in one of them has a unique neighbor in the other.
Given a set S ⊆ V (Q n ), S D (u) denotes the set S ∩ V (Q D (u)).
To engineer our inductive construction, we employ the following result on hypercube partitioning due to Wiener [15,Theorem 2.5], see also [16,Section 4] for the proof. Although it was originally stated for set systems, here we provide an equivalent formulation using the terminology introduced above.
Theorem 1 ( [15]). Let S be a set of vertices of Q n of size s with s ≥ 2n and d = A matching is a set of pairwise non-adjacent edges. Given a matching M , we use M to denote the set of all vertices incident with an edge of M . We say that a matching M in K(Q n ) is d-saturated if every short edge of dimension d not in M is adjacent to some edge of M .
Note that removing all edges of some fixed dimension d splits Q n into two (n − 1)-dimensional sub- The following properties of hypercubes shall be useful later.

Construction
The following construction is a refinement of Fink's method [8, Proof of Theorem 3] which was originally devised to extend perfect matchings in hypercubes to Hamiltonian cycles.
. Select a perfect matching P 0 in the subgraph of K(Q 0 ) induced by u 1 , u 2 , . . . , u k .

5.
The removal of edges of P 0 breaks C 0 into pairwise disjoint paths, and we can without loss of generality assume that these are paths P u1u2 , P u3u4 , . . . , P u k−1 u k .
Note that Construction 4 is non-deterministic in the sense that edges selected in steps 2 and 3 as well as a matching P 0 and cycles C 0 and C 1 formed in steps 4 and 5 may not be unique or may not even exist. If C 0 and C 1 are cycles of K(Q 0 ) and K(Q 1 ), respectively, defined in steps 4 and 5, where M i d , P i , S i , i ∈ {0, 1} used in their definition are obtained by some execution of Construction 4, then the pair (C 0 , C 1 ) is called a d-extension of the matching M in K(Q n ).
Proof: Referring to the notation of Construction 4, replace each edge v i v j ∈ P 1 in C 1 with the path v i + P uiuj + v j . This extends C 1 to a cycle C in K(Q n ) such that E(C) = M ∪ S 0 ∪ S 1 ∪ S 2 where S 2 consists of edges added in Step 2 or 3.

Initial step
The following application of Construction 4 shall be useful as an initial step for the inductive proof of our main result. Lemma 6. Every matching M in K(Q n ), |M | ≤ n ≥ 2, may be extended by short edges to a cycle of length at least 3 4 |V (Q n )|.

Proof:
We argue by induction on n. The case n = 2 may be verified by a direct inspection. Next assume that n > 2 and that M = ∅, for otherwise we may use an arbitrary Hamiltonian cycle of Q n .
Case A. n = 3 and for every d ∈ [3], either |M 2 d | = ∅, or |M 2 d | is odd and M is d-saturated. Recall that if M is d-saturated, Observation 2 says that |M | ≥ 2 + s/2 where s is the number of short edges of dimension d. Consequently, |M | = 2 would mean that M consists of two long edges, which together with n = 3 implies that |M 2 d | = 2 for some d, contrary to our assumption. Hence it must be the case that |M | = 3. If all these three edges are short, M extends to a Hamiltonian cycle by part (3) of Proposition 3. If at least one edge is long, then there is d ∈ [n] such that |M 2 d | > 1 and therefore, by our assumption, |M 2 d | = 3.

Hence we can assume that
. Since M is d-saturated, one of these edges, say u 1 v 1 , must have the property that v d 1 is not incident with any edge of M . Then P 0 = {u 1 v d 1 , u 2 u 3 } forms a perfect matching of K(Q 0 ) and therefore, by the induction hypothesis, it may be extended by short edges to a Hamiltonian cycle C 0 of Q 0 . Since Q 1 = Q {d} (1) is a cycle, there exists a path P u d Recall that M = ∅ and therefore there must be a d such that M is d -saturated. By Observation 2, |M | ≥ 4 + s/2 where s is the number of short edges of M of dimension d . Since we assume that |M | ≤ 4, it follows that |M | = 4 and every edge e ∈ M with d ∈ dim(e) is long. But then the are distinct edges e, e ∈ M such that dim(e) and dim(e ) share the same dimension d, which means that Hence we can without loss of generality assume that forms a perfect matching of K(Q 0 ). By part (2) of Proposition 3, P 0 may be extended by short edges to a Hamiltonian cycle C 0 of Q 0 . Since |E(C 0 ) \ E(P 0 )| = 4, there is an edge uv ∈ E(C 0 ) \ E(P 0 ) which is not incident with v d i for any i ∈ [3]. Replacing edges uv and u i v d i of C 0 with the paths u, u d , v d , v and u i , v i , v d i for all i ∈ [3], respectively, we extend C 0 to a cycle C of length We show that in this case there is a d ∈ [n] such that a d-extension of M exists. If 3 ≤ n ≤ 4, let d be such that either |M 2 d | is even and positive, or |M 2 d | is odd and M is not d-saturated. If n ≥ 5, select d ∈ [n] in the following way: If |M | < n, let d be an arbitrary dimension of some edge of M . If |M | = n and for each i ∈ [n] there is exactly one edge e ∈ M with i ∈ dim(e), then there must be two distinct short edges e 1 , e 2 ∈ M . Then use part (1) of Proposition 3 to select d in such a way that these edges belong to distinct subcubes Q 0 = Q {d} (0) and Q 1 = Q {d} (1). If none of the previous cases applies, d may be selected such that M contains at least two edges of dimension d.
Now verify the validity of all steps of Construction 4, using the notation introduced there. First inspect Step 2, i.e. the case when |M 2 d | is odd. If 3 ≤ n ≤ 4, then M is not d-saturated by the above choice of d. For n ≥ 5 we have 2|M | ≤ 2n < 2 n−1 , which means that by Observation 2, M is not d-saturated either. It follows that Step 2 may be safely performed. Since Step 3 is irrelevant, and hence it remains to verify the validity of Steps 4 and 5. Note that both M 0 d ∪ P 0 and M 1 d ∪ P 1 are matchings in K(Q 0 ) and K(Q 1 ), respectively, while our choice of d guarantees that their size does not exceed n − 1 unless n = 4 when it may be equal to n. By the induction hypothesis (or by part (2) of Proposition 3 in the case that n = 4), these matchings may be extended by short edges to cycles C i of lengths at least 3 4 |V (Q i )| for both i ∈ {0, 1}. The desired cycle of length at least 3 4 |V (Q n )| extending M by short edges then exists by Observation 5.

Results
We start with another application of Construction 4. This time, the assumption on matching size is replaced with a requirement of even distribution of its edges among 4-dimensional subcubes.
Note that in Case A, the subcubes Q D (u) and Q D (v) are adjacent by our assumption, and hence we can select d as the dimension of the edge uv. Otherwise d may be an arbitrary element ofD (in Case D the only one). Now go through the steps of Construction 4, using the notation introduced there. To perform Step 2 or 3, we need to find one or two non-adjacent short edges of dimension d, not intersecting any edge of M . To that end, select w ∈ {0, 1} n−4 such that • in Case C, w may be arbitrary, • in Case D, Q 5 is partitioned into subcubes Q D (w) and Q D (w d ).

Note that then
Hence we can always select It remains to verify the validity of Steps 4 and 5. Note that both M 0 d ∪P 0 and M 1 d ∪P 1 are matchings in K(Q 0 ) and K(Q 1 ), respectively, inheriting the required partition into subcubes of dimension four, where each subcube -in Cases A, B and C -contains no more than 7 vertices incident with edges of M . Moreover, the only subcubes with 6 or 7 such vertices might be • Q D (w) in Q 0 , and Q D (w d ) in Q 1 (Case C), without loss of generality assuming that Q D (w) lies in Q 0 = Q {d} (0). By the induction hypothesis, these matchings may be extended by short edges to cycles C i of lengths at least 3 4 |V (Q i )| for both i ∈ {0, 1}. Case D requires a special attention. Recall that in this case we had Q 5 partitioned into two subcubes such that |Q D (w)|, |Q D (w d )| ≤ 7. If |M 2 d | was odd, Step 2 could have increased this number to 8. If M 2 d was empty, both |Q D (w)| and |Q D (w d )| were even and therefore at most 6.
Step 3 would then increase this number to at most 8. Consequently, M 0 d ∪ P 0 and M 1 d ∪ P 1 in Steps 4 and 5 are matchings in 4dimensional subcubes K(Q 0 ) and K(Q 1 ), respectively, of sizes not exceeding 8/2 = 4. By Lemma 6, these matchings may be extended by short edges to cycles C i in K(Q i ) of lengths at least 3 4 |V (Q i )| for both i ∈ {0, 1}.
In all cases, the desired cycle of length at least 3 4 |V (Q n )| extending M by short edges exists by Observation 5.
It remains to use Theorem 1 to transform the assumptions of the previous lemma into an upper bound on the matching size.
Theorem 8. Let M be a matching in K(Q n ), n ≥ 2, such that | M | < n 2 6 + n 2 + 1. Then there is a set S ⊆ E(Q n ) such that M ∪ S forms a cycle in K(Q n ) of length at least 3 4 |V (Q n )|.
Proof: For n ≤ 8 the statement of the theorem follows from Lemma 6. For n > 8 select a set S such that M ⊆ S ⊆ V (Q n ) and |S| = n 2 6 + n 2 . Then Since the set of matchings in K(Q n ) includes all matchings in Q n , as a corollary we obtain the main result of this paper.
Corollary 9. Every matching M in Q n such that n ≥ 2 and |M | < n 2 12 + n 4 + 1 2 can be extended to a cycle of length at least 3 4 |V (Q n )|.