Parity balance of the i -th dimension edges in Hamiltonian cycles of the hypercube
PParity balance of the i -th dimension edges in Hamiltonian cyclesof the hypercube Feli´u Sagols ∗ , Guillermo Morales-Luna † April 1, 2019
Abstract
Let n ≥ i ∈ { , . . . , n − } . An i -th dimension edge in the n -dimensionalhypercube Q n is an edge v v such that v , v differ just at their i -th entries. The parity of an i -thdimension edge v v is the number of 1’s modulus 2 of any of its vertex ignoring the i -th entry. Weprove that the number of i -th dimension edges appearing in a given Hamiltonian cycle of Q n withparity zero coincides with the number of edges with parity one. As an application of this resultit is introduced and explored the conjecture of the inscribed squares in Hamiltonian cycles of thehypercube: Any Hamiltonian cycle in Q n contains two opposite edges in a 4-cycle. We prove thisconjecture for n ≤
7, and for any Hamiltonian cycle containing more than 2 n − edges in the samedimension. This bound is finally improved considering the equi-independence number of Q n − , whichis a concept introduced in this paper for bipartite graphs. Keywords : Hypercube; Hamiltonian cycles; i -th dimension edges; equi-independence number;inscribed square conjecture; bipartite graphs. Let n ≥ n -dimensional hypercube , denoted Q n , is the graph having { , } n as set of vertexes, two of them being joined by an edge if they differ just in one of their entries. It iswell known that hypercubes are all Hamiltonian graphs. The canonical examples of Hamiltonian cyclesin the hypercube are the so called binary Gray codes [9].The study of structural properties of the Hamiltonian cycles of the hypercube have allowed the solutionof relevant problems. For instance, the proof of Kreweras’s conjecture by Fink in 2007 (see [6, 7]) aboutthe extensibility of any perfect matching on the hypercube allowed Feder and Subi in 2009 to find tightbounds on the number of different Hamiltonian cycles in the hypercube [5]; the same structural propertyallowed Gregor in 2009 to prove that perfect matchings on subcubes can be extended to Hamiltoniancycles of hypercubes [8]. Something similar has been done with Hamiltonian paths, for example Chenproved in 2006 (see [2]) that any path of length at most 2 n − Q n and he used this basic structural property to prove that, for n ≥ Q n is (2 n − n − P of length k , with 2 ≤ k ≤ n − k −
2) if and only if P contains two edges in the same dimension [2].Our objective here is to extend the structural knowledge about Hamiltonian cycles of the hypercubeby classifying permissible sets of edges in Hamiltonian cycles, in terms of the dimension to which theseedges belong and their parities (see Theorem 4.1).As an application we conjecture (see Conjecture 5.1) that any Hamiltonian cycle in the hypercubecontains the opposite edges of a 4-cycle, and we prove some particular instances of this conjecture (seeTheorem 4.1). In the 80’s Erd¨os [4] conjectured that any subgraph of Q n asymptotically containing atmost half edges of the whole Q n is 4-cycle free, recent advances on this conjecture appeared in [1] and [10].The substantial difference of Erd¨os conjecture and ours is that Erd¨os conjecture poses conditions on the ∗ Mathematics Department, CINVESTAV-IPN Mexico City. [email protected] † Computer Science Department, CINVESTAV-IPN Mexico City. [email protected] a r X i v : . [ m a t h . C O ] S e p aximum number of edges in a 4-cycle-free graph and our conjecture is about the existence of 4-cycleswith opposite edges in any Hamiltonian cycle of the hypercube.As a first approach to explore Conjecture 5.1 it is introduced the notion of i -th dimension graph whichin turn is isomorphic to Q n − and allows us to translate our decision problem (whether there is a 4-cyclewithin any Hamiltonian cycle h in the hypercube) into verifying that the maximum number of edges ina dimension in E ( h ) is greater than the independence number of Q n − (see Theorem 5.1). Then the lastbound is improved as the equi-independence number of Q n − (see Corollary 5.1). The equi-independencenumber of a bipartite graph G is the cardinality of the maximum independent set in G containing the samenumber of elements in each bipartition class; we prove in Theorem 5.3 that finding the equi-independencenumber of a graph is polynomial time reducible to the independence number computation problem.The outline of the paper is the following: In Section 2 we recall very basic notions of Graph Theoryand we introduce some notation, in Section 3 we introduce and study i -th dimension graphs. In Section 4we introduce the notion of chromatic vector and some other operators defined over Hamiltonian cycles inthe hypercube, and we prove Theorem 4.1 which is the main result in this paper. In Section 5 we stateand discuss the inscribed square conjecture, we introduce the equi-independence concept for bipartitegraphs and we report our final progress on this conjecture. In the conclusions, we suggest additionalapplications to Theorem 4.1 and we pose a list of open conjectures and problems. Let G = ( V, E ) be graph with set of nodes V and set of edges E . Let us recall the following elementarynotions. A k -cycle in G is a sequence of pairwise different nodes v v . . . v k − such that for each index i , v i v i +1 is an edge in E (index addition is taken modulus k ). A Hamiltonian cycle is a k -cycle, where k = card( V ) is the order of the graph. The graph G is Hamiltonian if it possesses a Hamiltoniancycle. A non-empty set I ⊂ V is independent if no pair of different elements in I is an edge: ∀ u, v ∈ I [ u (cid:54) = v ⇒ uv (cid:54)∈ E ]. A maximal independent set is an independent set which is maximal with respect toset-inclusion. The independence number α ( G ) of G is the number of vertexes in a largest independentset: α ( G ) = max { ν | ∃ I ⊂ V : I independent & card( I ) = ν } . Any independent set consisting of α ( G )vertexes is called a maximum independent set .Let n ≥ E ( Q n ) consists of pairs v ( v + e i ), where e i is the i -thcanonical basic vector. Clearly, v ( v + e i ) = u ( u + e j ) if and only if i = j and either u = v or u = v + e i (addition is integer addition modulus 2).Any 4-cycle in Q n has thus the form v ( v + e i )( v + e i + e j )( v + e j ), with v ∈ V ( Q n ) and 0 ≤ i < j ≤ n − Q n is thus 2 n and its independence number is α ( Q n ) = 2 n − .The hypercube Q n is Hamiltonian and the binary Gray code [9], Gr n , is a Hamiltonian cycle. As asequence, this code is determined recursively by the following recurrence:Gr = [0 , , Gr n = join(0 ∗ Gr n − , ∗ rev(Gr n − ))(join and rev are respectively list concatenation and list reversing, ∗ is a prepend map: b ∗ list prependsthe bit b to each entry at the list). In general, we will follow the notions and notations in Diestel’stextbook [3]. Let n ≥ i be an integer in { , . . . , n − } . An i -th dimension edge in Q n is and edgeof the form u ( u + e i ) where u ∈ V ( Q n ).For each i ∈ { , . . . , n − } let D in be the i - dimension graph whose vertexes are the i -th dimensionedges of Q n , and whose edges are the pairs of i -th dimension edges forming a 4-cycle within Q n : v ( v + e i ) u ( u + e i ) is an edge in D in ⇐⇒ ∃ j (cid:54) = i : u ( u + e i ) = ( v + e j )( v + e j + e i ) . (1) Theorem 3.1.
Let n ≥ be an integer. For each i ∈ { , . . . , n − } the graph D in is isomorphic to Q n − . roof: The map f in : V ( D in ) → V ( Q n − ), v ( v + e i ) (cid:55)→ f in ( v ( v + e i )) = ρ in ( v ), where ρ in is deletion ofthe i -th entry, is a graph isomorphism.Hence, being all isomorphic to Q n − , the i -th dimension graphs D in are pairwise isomorphic and each D in is bipartite.Let par n : Q n → { , } be the map such that par n ( v ) is the parity of the Hamming weight (numberof entries equal to 1) of the vector v ∈ Q n , and for any index i ∈ { , . . . , n − } let par in = par n − ◦ ρ in where ρ in is the map that suppresses the i -th entry at any n -dimensional vector. In other words, par in ( v )is the parity of the vector resulting by suppressing the i -th entry in v . We have that the bipartitionclasses of D in are realized as: v v , v v in the same bipartition class ⇐⇒ par n − ◦ f in ( v v ) = par n − ◦ f in ( v v ) . Or equivalently, v ( v + e i ) , u ( u + e i ) in the same bipartition class ⇐⇒ par in ( v ) = par in ( u ) . (2)According to the common value at the right side of relation (2) we will refer to the partition classes as0- and 1- bipartition classes of D in .Let h = h · · · h n − be a Hamiltonian cycle of Q n . For each ι ∈ { , . . . , n − } let us associate thedimension i of the edge h ι h ι +1 as a color of the starting vertex (in the sense of the cycle) h ι . Let usdenote χ h : V ( Q n ) → { , . . . , n − } this vertex coloring induced by the Hamiltonian cycle h .Let c ( h ) = ( c i ) n − i =0 ∈ N n be the vector such that for each i , c i is the number of vertexes colored i bythe Hamiltonian cycle h . The vector c ( h ) is called the chromatic vector of h . c : { Hamiltonian cycles } → N n is a well defined map. A full characterization of the image C ⊂ N n ofthis map is out of the scope of this paper. However, some necessary conditions for chromatic vectors areasserted at the following Lemma 3.1.
Let n ≥ be an integer and let c ∈ C be the chromatic vector of some Hamiltonian cycle h , c = c ( h ) . Then:1. All entries of c are even integers.2. No entry at c is zero.3. The sum of the entries of c is equal to n .4. The greatest entry in c is lower or equal than n − .5. The lowest entry in c is greater or equal than .6. Any permutation of c is the chromatic vector of a Hamiltonian cycle of Q n .Proof: The first assertion follows from the fact that on the hypercube Q n , (cid:80) v ∈ Q n v = 0 ∈ V ( Q n ) (sumis addition modulus 2). The second because if c i = 0, for some i , then the nodes at the Hamiltonian h either all lie at the semi-space v i = 0 or at the opposite semi-space v i = 1, which is not possible. Thirdassertion holds because each vertex has assigned a color. Fourth assertion follows from the fact that noconsecutive pair of edges in h can be parallel, or, in other words, if h j h j +1 has the same direction as h j +1 h j +2 , for some j ∈ { , . . . , n − } , then necessarily h j = h j +2 but that is impossible. Fifth assertionis a consequence of the first and the second. For the last assertion, let π : { , . . . , n − } → { , . . . , n − } be a permutation, then c ( π ◦ h ) = π ( c ( h )) and π ◦ h is a Hamiltonian cycle.Let h = h · · · h n − be a Hamiltonian cycle in the hypercube and let i ∈ { , . . . , n − } . By rotatingand reverting, if necessary, the indexes in { , . . . , n − } , we may assume that i is the color assigned to h and that h i = 0 (indeed, h h is an i -th dimension edge). Let us define the following operators: inverse image index list. λ i ( h ) = [ j , j , . . . , j c i − ] ( j = 0): the list of indexes colored i at theHamiltonian h . This is the list of indexes (cid:96) ∈ { , . . . , n − } such that χ h ( h (cid:96) ) = i . In other words,it is the list of indexes in h where an i -th dimension edge starts,3 nverse image lists. η i ( h ) = [ h j , h j , . . . , h j ci − ], and ξ i ( h ) = [ h j h j +1 , h j h j +1 , . . . , h j ci − h j ci − +1 ].The list η i ( h ) consists of the i -th colored vertexes at Q n , and the list ξ i ( h ) consists of the i -thdimension edges at Q n .Let us study the maps λ i , η i , ξ i . Let n ≥ n -dimensional hypercube Q n is bipartite and its bipartite classes are thecollections of vertexes of odd and even parity: P n = par − n (0), P n = par − n (1). Let h = h · · · h n − bea Hamiltonian cycle in the n -dimensional hypercube Q n . Remark.
The sequence par n ◦ h consists of alternating values 0 and 1. Theorem 4.1.
Let n ≥ be an integer, let i ∈ { , . . . , n − } and let h be a Hamiltonian cycle in Q n .Then half of the i -th dimension edges appearing in h , namely at the list ξ i ( h ) , lie in the -bipartition classof D in and the other half in the -bipartition class.Proof: The first statement at Lemma 3.1 asserts that c i = | ξ i ( h ) | is indeed an even integer. Without lossof generality we may assume that h h is an i -th dimension edge and the i -th entry of the starting vertex h has value 0, i.e. h i = 0.For a bit b ∈ { , } , let Q bin be the subgraph of Q n induced by those vertexes with value b at their i -th entry. Both Q in and Q in are isomorphic to Q n − and they are bipartite as well.Let λ i ( h ) = [ j , j , . . . , j c i − ] ( j = 0) be the index list corresponding to i -colored nodes at h . Fromrelation (2) we have ∀ k < c i , b ∈ { , } [ par in ( h j k ) = par in ( h j k +1 ) = b ⇐⇒ h j k h j k +1 ∈ b -bipartition class] . (3)Let us consider the parity list [ par in ( h j k )] c i − k =0 . By relation (3), the theorem will be proved by showingthat half values in this list are 0 and half are 1.Since h i = 0, for any even k ∈ { , , . . . , c i − } the path H k = h j k +1 · · · h j k +1 lies entirely in Q in .Let V ( H k ) be the collection of vertexes appearing in H k . Indeed, { V ( H k ) } ci − k =0 is a partition of V ( Q in ) . (4)Similarly, { V ( H k +1 ) } ci − k =0 is a partition of V ( Q in ) . (5)Figure 1 displays a diagram of this situation. According to relation (3), the following equivalences hold: par in ( h j k +1 ) (cid:54) = par in ( h j k +1 ) ⇐⇒ the length of H k is odd, (6) par in ( h j k +1 ) = par in ( h j k +1 ) ⇐⇒ the length of H k is even. (7)If the inequation at the left side of (6) holds then the endpoints of H k have different parities and theydoes not introduce any imbalance on the number of zeros and ones in [ par in ( h j k )] c i − k =0 (recall that bydefinition par in ( h j k +1 ) = par in ( h j k )). If the equation at the left side of (7) holds with value b ∈ { , } forsome even number k = k , then h k has an odd number of vertexes and in consequence one vertex less inthe b -bipartition class of Q in than in the b -bipartition class. Since relation (4) is true this deficit must becompensated with the existence of some even value k = k such that par in ( h j k +1 ) = par in ( h j k ) = b .In other words card (cid:0) { k ∈ { , , . . . , c i − }| par in ( h j k +1 ) = 0 = par in ( h j k +1 } (cid:1) =card (cid:0) { k ∈ { , , . . . , c i − }| par in ( h j k +1 ) = 1 = par in ( h j k +1 } (cid:1) Consequently [ par in ( hj k )] c i − k =0 shall have the same number of 0’s and 1’s.The above result entails some necessary conditions for a vector to be of the form c ( h ) for someHamiltonian cycle h .For a Hamiltonian cycle h on Q n and an index i ∈ { , . . . , n − } let us define:4 j h j h j h j h j h j c −1i dimension edges i−th H H Q Q H H H c −2i H c −1i Figure 1: The traces of a Hamiltonian cycle on the parallel semi-hypercubes with constant i -th entry. chromatic-segments list. δ i ( h ) = [ j − j , j − j , . . . , n − j c i − ]: the difference among the right shiftof λ ( h ) and λ ( h ) itself, parity list. β i ( h ) = [ b , b , . . . , b c i − ]: bit list defined recursively as follows: b = par in ( h ) & ∀ k ∈ { , . . . , c i − } : b k = ( b k − + j k − j k − + 1) mod 2 . Corollary 4.1.
Let n ≥ be an integer and i ∈ { , . . . , n − } , and let h be a Hamiltonian cycle on Q n .Then the i -th parity list of h , β i ( h ) , is a balanced list, i.e. the number of and values coincide.Proof: A simple induction proves that β i ( h ) = [ par in ( h j k )] c i − k =0 . The conclusion follows from the proof ofTheorem 4.1. Corollary 4.2.
Let n ≥ be an integer and i ∈ { , . . . , n − } . The addition of the even numberedentries at the chromatic-segments list δ i ( h ) coincides with the addition of its odd numbered entries andboth are equal to n − : ci − (cid:88) k =0 ( δ i ( h )) k = 2 n − = ci − (cid:88) k =0 ( δ i ( h )) k +1 (8) Proof:
For any k ∈ { , . . . , c i − } the path H k in the proof of Theorem 4.1 has δ k ( h ) vertexes. The first(resp. last) expression at relation (8) corresponds to the sum of the lengths of paths h k with even (resp.odd) index and the result follows from relations (4) and (5). We introduce here an application of Theorem 4.1. We will establish sufficient conditions over Hamiltoniancycles, in terms of their chromatic vectors, to guarantee that there are 4-cycles within the Hamiltoniancycles.Let n ≥ h = h · · · h n − be a Hamiltonian cycle of Q n . Let v v v v be a4-cycle in the hypercube Q n . Let us introduce the following definitions:1. v v v v is a straight square within h if there exist indexes i, j ∈ { , . . . , n − } such that h i h i +1 h j h j +1 = v v v v . v v v v v v v v v v v v v v v v a)b) Figure 2: Hamiltonian cycles possessing squares. (a) Straight squares. (b) Twisted squares.2. v v v v is a twisted square in h if there exist indexes i, j ∈ { , . . . , n − } such that h i h i +1 h j h j +1 = v v v v .
3. We will say that the edges v v and v v are the rims and the edges v v and v v are the rays .In figure 2 we illustrate those notions, the rays are displayed as dashed lines, the rims as continuouslines since they actually “are part of” the Hamiltonian cycle. We will say that the Hamiltonian cycle h contains a square , or that a square is inscribed in h , if a straight or twisted square is contained in h . A square free Hamiltonian cycle contains no squares. Conjecture 5.1 (Inscribed squares in Hamiltonian cycles.) . Let n ≥ be an integer. No Hamiltoniancycle of Q n is square free, or equivalently any Hamiltonian cycle has inscribed a square. Since any 4-cycle in Q n contains two alternating dimensions, the rims (or rays) in each square inscribedin a Hamiltonian cycle of Q n belong to the same dimension.Now let us establish some conditions in order to have edges of the same dimension in a square containedwithin a Hamiltonian cycle of Q n .From the construction of graph D in , specially the edge definition at relation (1), and the Theorem 3.1,if the chromatic vector of some Hamiltonian cycle h of Q n contains entries greater than the independencenumber of Q n − then h must contain a square. Since α ( Q n − ) = 2 n − the following theorem results. Theorem 5.1.
Let n ≥ be an integer, and let h be a Hamiltonian cycle of Q n . If some entry i in thechromatic vector c ( h ) is greater than n − then h contains a square whose rims are i -th dimension edges. Theorem 5.1 is a general result aiming to prove Conjecture 5.1 for Hamiltonian cycles whose chromaticvectors have entries bigger than 2 n − . Let us reduce further this lower bound.Let G be a bipartite graph. The equi-independence number of G , denoted α = ( G ), is the cardinalityof the maximum independent set in G having half of its vertexes in one bipartition class of G and half inthe other.According to the proofs of Theorem 5.1 and Theorem 4.1 we have: Corollary 5.1.
Let n ≥ be an integer, and let h be a Hamiltonian cycle of Q n . If some entry c i in c ( h ) is greater than α = ( Q n − ) then h contains a square whose rims are i -th dimension edges. Let us estimate a lower bound for the equi-independence number of the n -dimensional hypercube. Theorem 5.2.
Let n ≥ be an integer. Then α = ( Q n ) ≥ n − .Proof : For any two bits b , b ∈ { , } let Q nb b be the subgraph of Q n induced by the vertexes in V ( Q n )having values b and b at their first two coordinates. The resulting four graphs are isomorphic to Q n − .6et b ∈ { , } , and let I b = V ( Q nbb ) ∩ ( b -bipartition class of Q n ). No vertex in I is adjacent to avertex of I and in fact I = I ∪ I is an independent set of Q n . Moreover, each vertex in V ( Q nbb ) notcontained in I is adjacent to one vertex in Q nbb , and each vertex in Q n (or in Q n ) is adjacent to avertex in I (or a vertex in I .)In order to illustrate the last claim let v = (0 , , b , . . . , b n − ) be an arbitrary vertex in Q n, , , if par n ( b , . . . , b n − ) = 0 then v is adjacent to (0 , , b , . . . , b n − ) which is in I , otherwise v is adjacent to(1 , , b , . . . , b n − ) which is in I .The final conclusion is that I is a maximal independent set in Q n . Since card( I ) = card( I ) = 2 n − and card( I ) = card( I ) + card( I ) = 2 n − + 2 n − = 2 n − the Theorem follows.Now let us check that finding the equi-independence number of a bipartite graph G is not moredifficult than evaluating the independence number of a graph G (cid:48) derived from G . Theorem 5.3.
The problem of finding the equi-independence number of a bipartite graph is polynomially-time reducible to the problem of finding the independence number of a graph.Proof : Let G be a bipartite graph, and let V , V be the bipartition classes of G . The graph G (cid:48) = ( V (cid:48) , E (cid:48) ),with set of vertexes V (cid:48) = { ( v , v ) ∈ V × V | ( v , v ) / ∈ E ( G ) } and set of edges E (cid:48) = { (( v , v ) , ( v (cid:48) , v (cid:48) )) ∈ V ( G (cid:48) ) × V ( G (cid:48) ) | ( v , v (cid:48) ) ∈ E ( G ) or ( v , v (cid:48) ) ∈ E ( G ) or v = v (cid:48) or v = v (cid:48) } ) , has an independent set I (cid:48) if and only if the set I = { v ∈ V ( G ) | v is an element in some vertex of I (cid:48) } is anequi-independent set of G . In this construction card( I ) = 2 card( I (cid:48) ). Since G (cid:48) can be built in polynomialtime, the proposition follows. .Theorem 5.3 was used to compute the equi-independence number of the hypercubes of dimensionthree to seven. The results are summarized in Table 1. Vertexes in the maximal equi-independence setsin Table 1 are coded in binary. The equi-independence numbers for n equals to four and six reach thelower bound in Theorem 5.2 and so this bound is tight. The graph Q (cid:48) n grows very fast as n is increased,thus for n = 8 the corresponding value is not included.Table 1: Equi-independence number of Q n for small values of n .n α = ( Q n ) A maximal equi-independent set | V ( Q (cid:48) n ) | | E ( Q (cid:48) n ) | {
0, 7 } {
0, 7, 9, 14 }
32 4485 10 {
0, 7, 9, 19, 10, 21, 12, 22, 24, 31 }
176 97206 16 {
0, 7, 9, 19, 33, 21, 10, 22, 34, 28, 882 13753636, 56, 43, 31, 45, 55 } {
0, 7, 9, 19, 33, 67, 10, 21, 34, 69, 3648 157718412, 81, 36, 22, 24, 70, 40, 82, 72, 84,48, 31, 96, 47, 57, 79, 105, 55, 58, 87,106, 103, 60, 91, 108, 115, 120, 93, 117, 126 } The study on the inscribed square conjecture in the Hamiltonian cycles of the hypercube concludesas follows.
Theorem 5.4.
Conjecture 5.1 holds for ≤ n ≤ .Proof : From Table 1 we know that α = ( Q ) = 16. It is impossible that the chromatic vector of aHamiltonian cycle of Q have all its entries lower or equal that 16; otherwise the sum of the seven entrieswould be at most 16 · n = 7. Theproof is analogous for values of n lower than 7.Computer experimentation suggests that the inscribed square conjecture in Hamiltonian cycles is truefor any value of n . 7 Conclusions
Theorem 4.1 entails several applications, e.g., in [6] it is posed the following question: The partialmatchings in Q n can be extended to Hamiltonian cycles? We are able to submit a partial answer tothis question: if the partial matching contains edges in the same dimension that violate the equilibriumcondition of the Theorem 4.1, and it is not allowed to incorporate more edges in the same dimension,then no such extension exists. Other application of the Theorem 4.1 consists in a pruning strategy togenerate exhaustively all the Hamiltonian cycles of Q n .On the other hand several problems and conjectures have remained open, and their study couldreveal important structural properties of Q n . In the following presentation we will refer to the operatorsintroduced at the end of section 3.As conjectures, besides the already stated Conjecture 5.1, let us state the following: • Let n ≥ and let k be an even integer with ≤ k ≤ n − . A vector v ∈ Q k is the parity vectorof a Hamiltonian cycle of Q n , i.e. v = β i ( h ) for some i and Hamiltonian cycle h , if and only if itcontains the same number of ’s and ’s. In the Theorem 4.1, the “only if” part of this conjecture was proved. • The lower bound in Theorem 5.2 is reached for all even values of n . As open problems, let us state the following: • Characterize the chromatic vectors of Hamiltonian cycles of the hypercube.
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