aa r X i v : . [ m a t h . C O ] N ov ON k -NEIGHBOR SEPARATED PERMUTATIONS ISTV ´AN KOV ´ACS AND DANIEL SOLT´ESZ
Abstract.
Two permutations of [ n ] = { , . . . n } are k -neighbor separated if there are two elementsthat are neighbors in one of the permutations and that are separated by exactly k − k -neighbor separated permutations of [ n ]be denoted by P ( n, k ). In a previous paper, the authors have determined P ( n,
3) for every n , answeringa question of K¨orner, Messuti and Simonyi affirmatively. In this paper we prove that for every fixedpositive integer ℓ , P ( n, ℓ + 1) = 2 n − o ( n ) . We conjecture that for every fixed even k , P ( n, k ) = 2 n − o ( n ) . We also show that this conjecture isasymptotically true in the following senselim k →∞ lim n →∞ n p P ( n, k ) = 2 . Finally, we show that for even n , P ( n, n ) = 3 n/ Introduction
There are numerous results concerning the maximum size of a family of permutations pairwisesatisfying some prescribed relation, see [3, 5, 6, 9, 11, 12, 15, 16]. There is a natural correspondencebetween permutations of n elements and Hamiltonian paths in the complete graph K n . Let G and G be two graphs on the same vertex set, we say that G is their union if V ( G ) = V ( G ) = V ( G )and E ( G ) = E ( G ) ∪ E ( G ) . K¨orner, Messuti and Simonyi made the following observation.
Proposition 1.1 ([17]) . The maximal number of Hamiltonian paths in the complete graph such thatevery pairwise union contains an odd cycle is equal to the number of balanced bipartitions of the vertexset. That is, on n + 1 vertices their maximal number is (cid:0) n +1 n (cid:1) and on n vertices it is (cid:0) nn (cid:1) = (cid:0) n − n (cid:1) . The upper bound follows by observing that a Hamiltonian path is a bipartite graph with a balancedbipartition. The union of two paths with the same bipartition is a bipartite graph which clearlycannot contain any odd cycle. On the other hand, if we choose a unique Hamiltonian path for everybalanced bipartition, the resulting family satisfies our condition. In [17] the authors asked whether theanswer remains the same if we ask for a triangle instead of an odd cycle. This question was answeredaffirmatively.
Theorem 1.2 ([14]) . The maximum number of Hamiltonian paths in the complete graph K n such thatevery pairwise union contains a triangle is equal to the number of balanced bipartitions of [ n ] . There are two natural ways to generalize this problem. We can consider it a problem for Hamiltonianpaths and ask for a k -cycle instead of a triangle. In this case, when k is even there are constructionsof size larger than exponential, see [8]. In this paper we take a different approach and we formulatethe problem in the language of permutations. The research of the first author was supported by National Research, Development and Innovation Office NKFIH,K-111827.The research of the second author was supported by the Hungarian Foundation for Scientific Research Grant (OTKA)No. 108947 and by the National Research, Development and Innovation Office NKFIH, No. K-120706. efinition 1.3. We say that two permutations of [ n ] = { , . . . n } are k -neighbor separated if there aretwo elements that are neighbors in one permutations and in the other permutation they are separatedby exactly k − elements. Let the maximal number of pairwise k -neighbor separated permutations of [ n ] be denoted by P ( n, k ) and let P ( k ) := lim n →∞ n p P ( n, k ) . Two permutations are 2-neighbor different if and only if their corresponding Hamiltonian pathsshare an edge. Determining P ( n,
2) is a significantly different task than determining P ( n, k ), for any k >
2. The reason for this is that it is that when k = 2 we are looking for similar Hamiltonian pathsinstead of different ones. Therefore the problem resembles Erd˝os-Ko-Rado like intersection problems.We will show that P ( n,
2) is the number of Hamiltonian paths containing a fixed edge. For the upperbound we will use Katona’s cycle method. We postpone this proof to Section 2.Two permutations are 3-neighbor separated if and only if their corresponding Hamiltonian pathsform a triangle in their union. Note that in the case of P ( n, P ( n, P ( n,
3) = (cid:0) n ⌊ n ⌋ (cid:1) when n ≡ (cid:0) n ⌊ n ⌋ (cid:1) when n ≡ . From Theorem 1.2 it also follows that P (3) = 2. For k > k -neighbor separated is stronger than the requirement that the corresponding Hamiltonian paths forma k -cycle. The authors conjecture the following. Conjecture 1.4.
For every integer < k, wehaveP ( k ) ≥ . Note that from Conjecture 1.4 it follows that for all positive integers k , P (2 k + 1) = 2 by thefact that the number of pairwise 2 k -separated permutations is at most the number of C k +1 -differentHamiltonian paths. But the number of these Hamiltonian paths is at most the number of balancedbipartitions of [ n ] by the upper bound in Proposition 1.1. For even values of k , the authors are morecautious, as in these cases sometimes there are larger constructions (found by computer) than thenumber of balanced bipartitions of the ground set, although these constructions are still smaller than2 n . The main result of the present paper is that Conjecture 1.4 holds for infinitely many k . Theorem 1.5.
For every positive integer ℓ , P (2 ℓ + 1) = 2 . Corollary 1.6.
For every positive integer ℓ , the maximal number of Hamiltonian paths of K n whereevery pairwise union contains a C ℓ +1 is n − o ( n ) . Proof.
The construction follows from Theorem 1.5 and for the upper bound, the proof of the upperbound of Proposition 1.1 applies verbatim. (cid:3)
We also prove upper bounds to P ( n, k ) for every k . Theorem 1.7.
For fixed < k we have P ( n, k ) ≤ ( n when k ≡ H ( k − k , k − k , k ) n when k ≡ where H ( x, y, z ) is the entropy function. From Theorem 1.7 it follows that lim sup k →∞ P ( k ) = 2. We also prove that lim inf k →∞ P ( k ) = 2 whichresults in lim k →∞ P ( k ) = 2 . We also investigate P ( n, n ). heorem 1.8. For every positive integer n the following holds P ( n, n ) = n when n ≡ (cid:4) n (cid:5) − ≤ P ( n, n ) ≤ (cid:4) n (cid:5) when n ≡ . A quick investigation by computer shows that the value of P (3 ,
3) and P (5 ,
5) is equal to the corre-sponding lower bound of Theorem 1.8, but the value of P (7 ,
7) attains the upper bound.The paper is organized as follows. In Section 3 we prove Theorem 1.5. In Section 4 we elaborateon the connection between the problem of determining P ( n, k ) and some Bollob´as-type questions. InSection 5 we prove that lim k →∞ P ( k ) = 2. In Section 6 we prove Theorem 1.8, followed by concludingremarks and some open questions. 2. When k = 2In this short section we determine the exact value of P ( n,
2) for all n . We use the following equivalentformulation: Two permutations are 2-neighbor different if and only if their corresponding Hamiltonianpaths share an edge. For the upper bounds we will use the following claims. Claim 2.1.
When n is even, the edges of the complete graph K n can be decomposed into edge disjointHamiltonian paths.Proof. Let us refer to the vertices of K n as { , , . . . , n } . Let F consist of the Hamiltonian path { , n, , n − , . . . , n/ } plus its first n/ − n/ F are pairwise edge disjoint by arranging the vertices { , , . . . , n } into a regular n -gon. Iftwo Hamiltonian path shares an edge that edge must have the same (euclidean) length in both paths.But edges of the same length come in antipodal pairs in the Hamiltonian paths of F (except the singleedge of maximal length in each path). Therefore rotation by at most n/ − (cid:3) Claim 2.2.
When n is odd, there is a set F of Hamiltonian paths of K n with size |F | = n . And theintersection graph G of F (the vertices are the Hamiltonian paths, two vertices are adjacent if thepaths share an edge) is a cycle of length n .Proof. Let us refer to the vertices of K n as { , , . . . , n } . Let M be the matching { (2 , n ) , (3 , n − , . . . , (( n + 1) / , ( n + 3) / } . For every i ∈ { , n } let us denote the rotated versions of M by M i := { (2 + ( i − , n + ( i − , (3 + ( i − , n − i − , . . . , (( n + 1) / i − , ( n + 3) / i − } where everything is understood modulo n . Let M := { M , . . . M n } . Observe that the matchings in M are pairwise edge disjoint. The union of the matchings H = M ∪ M ( n +1) / is a Hamiltonian path.For each 2 ≤ i ≤ n let H i := M i ∪ M ( n − / i − . We claim that the set F := { H , . . . H n } satisfiesthe conditions of our claim. The H i are Hamiltonian paths since for every 2 ≤ i ≤ n , H i is just therotated version of H . If two Hamiltonian paths in F share an edge, then they share a matching in M since the matchings in M were edge disjoint. Thus every Hamiltonian path H i in F corresponds to aset of two matchings, that are { M i − , M ( n +1) / i − } and two Hamiltonian paths share an edge ifand only if their corresponding set of matchings intersect. Thus it is easy to see that, the intersectiongraph of the Hamiltonian paths in F is indeed a cycle of length n . (cid:3) Theorem 2.3.
For every positive integer n , we have P ( n,
2) = ( n − . Proof.
The lower bound ( n − ≤ P ( n,
2) follows from the observation that the number of Hamiltonianpaths of K n containing a fixed edge is ( n − n is even: By Claim 2.1, there is a decomposition of the edges of K n intoHamiltonian paths. Let us fix such a decomposition F . Let us permute the ground set under F : Forevery π permutation of the set [ n ], we define F π to be the Hamiltonian path decomposition of K n ob-tained from F by relabelling the vertices of the ground set from { , , . . . , n } to { π (1) , π (2) , . . . , π ( n ) } . et H be a family of Hamiltonian paths such that every pair of Hamiltonian paths in H have anedge in common. Since for every π , F π consists of edge disjoint Hamiltonian paths, clearly we have |H ∩ F π | ≤
1. Moreover, for every Hamiltonian path H (not necessarily in H ), there are exactly n permutations π , . . . , π n such that for all i ∈ { , . . . , n } , H ∈ F π i (Every Hamiltonian path in F canbe relabelled to H in exactly two ways). Therefore n |H| ≤ X π ∈ S n |H ∩ F π | ≤ n ! . Thus the proof of the case where n is even is complete.The upper bound when n is odd: We proceed similarly to the case when n is even. Let F be a familyof Hamiltonian paths as in Claim 2.2. for every permutation π , we define F π to be the Hamiltonian pathdecomposition of K n obtained from F by relabelling the vertices of the ground set from { , , . . . , n } to { π (1) , π (2) , . . . , π ( n ) } . Let H be a family of Hamiltonian paths such that every pair of Hamiltonianpaths in H have an edge in common. By Claim 2.2, for every permutation π , we have |H ∩ F π | ≤ H (not necessarily in H ), there are exactly 2 n permutations π , . . . , π n such that for all i ∈ { , . . . , n } , H ∈ F π i (Every Hamiltonian path in F can be relabelled to H inexactly two ways). Therefore 2 n |H| ≤ X π ∈ S n |H ∩ F π | ≤ n !and the proof is complete. (cid:3) Remark.
Theorem 2.3 was also proven independently by Casey Tompkins, [4].3.
The lower bound for k = 2 ℓ + 1We say that two Hamiltonian paths are k -neighbor separated if their union contains a k -cycle, suchthat one of them has at least k edges in that cycle. It is easy to see that two permutations are k -neighbor separated if and only if their corresponding Hamiltonian paths are k -neighbor separated. Wewill construct Hamiltonian paths instead of permutations. The main advantage of this is that we candraw helpful figures. We construct a suitable system of Hamiltonian paths in two steps. First we buildan appropriate set of so-called “labelled graphs”, then we construct numerous Hamiltonian paths fromeach labelled graph.3.1. Labelled graphs.Definition 3.1.
A labelled graph is a graph where each edge gets a label from { a, b } . We will be interested in disconnected labelled graphs where every connected component is a labelledgrid that is defined as follows.
Definition 3.2.
A labelled grid of width w and height h is a graph on wh vertices of the form ( i, j ) where ≤ i ≤ w and ≤ j ≤ h . Two vertices are connected by an edge of label a if they differ only intheir first coordinate and the difference there equals one. Similarly, two vertices are connected by anedge of label b if they differ only in their second coordinate and the difference there is one, see Figure3.1. Definition 3.3. A w -labelled graph is the disjoint union of an isolated vertex and some labelled gridsof width exactly w . The following construction describes, how to build Hamiltonian paths from w -weighed graphs. (Inthis paper we will build Hamiltonian paths only from w -labelled graphs for some integer 2 ≤ w .) igure 1. In our figures we draw the edges of label a as ordinary edges and the edgesof label b as dashed ones. Here we see five labelled grids, their width and height in a( w, h ) format from the left to right is: (1 , , (3 , , (1 , , (2 ,
2) and (3 , . Figure 2.
How the 0 − w = 3. Z-swapping construction.
Let 2 ≤ w and W be a w -labelled graph and let g be the number oflabelled grids of width w in W . We construct 2 g Hamiltonian paths from W as follows. Fix an order ofa components of W where the first component is the isolated vertex. The Hamiltonian paths that weconstruct will be indexed by 0 − g . Each Hamiltonian path starts at the isolatedvertex of W , and visits the labelled grids according to the fixed order. At the i -th labelled grid if the i -th element in the 0 − a of the labelled grid,and it visits the rows of the grid from the top to the bottom. Moreover if it started in the top rightcorner then it traverses every row from the right to the left, otherwise it traverses every row from leftto the right, see Figure 2. Claim 3.4.
The Z -swapping construction applied to a w -labelled graph with g labelled grids (notcounting the isolated vertex) produces g pairwise w + 1 -neighbor separated Hamiltonian paths.Proof. The number of Hamiltonian paths is immediate from the definition. For the w + 1-neighborseparatedness, consider two different Hamiltonian paths. These correspond to two 0 − he first difference of these sequences occurs at the i -th coordinate, then the Hamiltonian paths differat the i -th labelled grid. This means that the last vertex of the ( i − − i -th grid inone path, and to the top right in the other. Therefore the last vertex of the ( i − w vertices at the top of the i -th grid form a w + 1-cycle in the union of the two Hamiltonian paths andactually both Hamiltonian paths contain w edges from this cycle. Therefore the Hamiltonian pathsare w + 1-neighbor separated. (cid:3) Observe that when we build a Hamiltonian path H from a labelled graph using the Z -swappingconstruction, the vertices that are connected by an edge of label a in W are neighbors in H . Thevertices that are connected by an edge of label b in W will be separated by w − H ifthe edge was in a grid of width w . Suppose that we have two labelled graphs W , W that contain onlygrids of width w (plus an isolated vertex), and both contain an edge e which gets label a in W andlabel b in W . Observe that the Hamiltonian paths constructed by the Z -swapping construction from W and W altogether form a pairwise w + 1-neighbor separated family. This motivates the followingdefinition. Definition 3.5.
Two labelled graphs are compatible if they share an edge which has different labels inthe two labelled graphs.Remark.
Note that for two graphs to be compatible we do not assume anything about the actualstructure of the graphs, only that they share an edge with different labels. In Section 4 we discuss theconnection between the compatibility of labelled graphs and some Bollob´as-type problems.In this paper the main connection between labelled graphs and Hamiltonian paths is the Z -swappingconstruction. From now on we work with labelled graphs instead of Hamiltonian paths. Let us startto build a large family of 2-labelled graphs. We will start every subsequent building process with thisfamily. We will build labelled grids of width 2 using labelled graphs of width 1 as building blocks. Stepby step, we will construct larger and larger families of pairwise compatible labelled graphs. Howeverin the intermediate steps the labelled graphs are not yet 2-labelled graphs since they might containgrids of width 1.3.2. Merging operations.
We describe a procedure that takes a special labelled graph W as input,and produces two compatible labelled graphs W and W by merging some of the labelled grids in W in two different ways. Simple merging operations.
Suppose that we have a labelled graph W that contains two labelledgrids g and g of the same width and height. Let us denote the corner points of g by { a, b, c, d } andthe corner points of g by { e, f, g, h } , see Figure 3.From W we can construct two new labelled graphs W and W , that differ from W only in the way g and g are merged. We describe these mergingmethods as follows.(1) Adjoining. In W the two grids are merged in such a way that the resulting labelled grid istwo times as wide as the original ones, so we connect the rightmost points ( b − d side) of g tothe corresponding leftmost points ( e − g side) of g by edges of label a , see Figure 3.(2) Rotating down. W is obtained from W by “rotating down” the second labelled grid: thetwo grids are merged in such a way that the resulting labelled grid is two times as high as theoriginal ones, we connect the vertices on the c − d side to the vertices on the h − g side byedges of label b , see Figure 3.Now W and W are compatible (by the edge dg ). Moreover, both W and W are compatible withany other graph that was originally compatible with W , since they contain W as a subgraph withunchanged labels. ac bd eg fhW ac bd eg fh W ac bdhf ge Figure 3.
The two labelled grids with corners { a, b, c, d } and { e, f, g, h } of width 3and height 4 can be merged in two different ways, see W for Adjoining and W for Rotating down . The two resulting labelled graphs are compatible by the edge dg . Multiple merging operation.
Let 1 ≤ w, h, m be positive integers and W be an arbitrary labelledgraph. Let M be a set of labelled grids of W that consists of 2 m labelled grids of equal width w and equal height h . From the pair ( W, M ) the multiple merging operation produces (cid:0) m ⌊ m ⌋ (cid:1) pairwisecompatible labelled graphs as follows.We pair up the 2 m grids into m ordered pairs and we also fix an arbitrary ordering of these orderedpairs. We build a family of labelled graphs that is indexed by the subsets of [ m ] of size ⌊ m ⌋ . For sucha subset S ⊂ [ m ], the corresponding labelled graph is formed as follows. We apply a simple mergingoperation on each pair, for the i -th pair: If i ∈ S then we merge the i -th ordered pair of labelled gridsby Adjoining . If i / ∈ S , we merge by Rotating down . Note.
For every ordered pair of labelled grids, when we apply the simple merging operations we alwaysuse the same labelling of the corners of the grids with the labels a, b, c, d, e, f, g, h . This will be necessaryto ensure compatibility of the resulting graphs.The multiple merging operation indeed produces (cid:0) m ⌊ m ⌋ (cid:1) pairwise compatible labelled graphs, sincetwo such labelled graphs correspond to two different subsets S , S of [ m ]. If j ∈ S △ S , then the j -thordered pair is merged in different ways in these labelled graphs. This ensures compatibility as can beseen in Figure 3. All the labelled graphs that are built during this operation are isomorphic to eachother. Note.
Instead of using subsets S ⊂ [ m ] that are of size ⌊ m ⌋ , we can actually use every possible subsetin the multiple merging operation. The resulting labelled graphs will still be compatible and we get 2 m pairwise compatible labelled graphs, but they are not pairwise isomorphic any more. In this paper we nly prove asymptotic results and by using every subset we would only gain subexponential factors. Wechoose to give up these subexponential gains for the property that every labelled graph is isomorphicto each other since this property makes the proofs considerably simpler. In [14] we determined theexact number of pairwise 3-neighbor separated permutations using similar techniques. In that paper,there is a part of the building process that corresponds to multiple merging operation and there weused every possible subset.The multiple merging operation is a useful tool to produce many pairwise compatible labelled graphsfrom a single one. But after we apply the multiple merging operation to a pair ( W, M ), some of thenew labelled grids will have their width doubled, and some of them their height doubled. Recall thatafter we build a suitable family of labelled graphs we wish to use the Z -swapping construction. Forthis we not only need that our labelled graphs are pairwise compatible but that they contain gridsof the same width. The purpose of the next operation is exactly this, although instead of having thewidth of every labelled grid constant, we will only keep the width of the vast majority of the gridsconstant. Width doubling operation
Let W be a labelled graph and let X be an induced subgraph of W thatis the disjoint union of labelled grids of the same width w and height h . Let | V ( X ) | = x and g be thenumber of labelled grids of X , thus g = xwh . The width doubling operation applied to a pair ( W, X )produces a family of pairwise isomorphic, pairwise compatible labelled graphs F ( W, X ), where eachelement of F ( W, X ) contains g/ − O (log g ) labelled grids of width 2 w , and |F ( W, X ) | = 2 g/ − O (log( g )) .The family F ( W, X ) is built as follows.Let a be the largest positive even integer so that a ≤ g and let X be a subgraph of X thatcontains a labelled grids of width w and height h . We apply the multiple merging operation onthe pair ( W, X ). Let F be the resulting family that contains (cid:0) a / ⌊ a / ⌋ (cid:1) pairwise compatible labelledgraphs. The labelled graphs in F are pairwise isomorphic as they are the result of a multiple mergingoperation. Two types of new labelled grids are formed in the process: ⌊ a / ⌋ grids of width w andheight 2 h and ⌈ a / ⌉ grids of width 2 w and height h .We proceed by defining families of labelled graphs inductively. Suppose that F i is already defined andit contains pairwise compatible pairwise isomorphic labelled graphs and |F i | = Q i − j =0 (cid:0) a j / ⌊ a j / ⌋ (cid:1) . Supposethat { a j } ij =0 are already defined and a j is the largest positive even number so that a j ≤ ⌊ a j − / ⌋ .Also suppose that every labelled graph in F i contains at least ⌊ a i / ⌋ labelled grids of width w andheight 2 i h . For each labelled graph W ′ ∈ F i we will apply a multiple merging operation as follows. Let a i +1 be the largest even integer smaller than or equal to ⌊ a i / ⌋ and let X i = X i ( W ′ ) be the subgraphof W ′ that contains a i +1 grids of width w and height 2 i h . We apply the multiple merging operationto the pair ( W ′ , X i ) to get the family of labelled graphs F W ′ , F i +1 := [ W ′ ∈F i F W ′ . The family F i +1 consists of Q ij =0 (cid:0) a j / ⌊ a j / ⌋ (cid:1) pairwise compatible and pairwise isomorphic labelledgraphs. Every labelled graph in F i +1 contains at least ⌊ a i +1 / ⌋ labelled grids of width w and height2 i +1 h . Since a > a > . . . by definition and each a i is a positive even number, there must be a lastelement in the sequence { a , a , . . . , a z } .We say that F ( W, X ) := F z is the output of the width doubling operation! Note.
At every multiple merging operation, we choose a i labelled grids of a certain height and widthwhere a i must be even. If the original number of those labelled grids were odd, there is still a “leftover”grid of that dimension in the next family and no further operations use this grid. Therefore after awidth doubling operation, there might be a small number (at most z ) of grids that have their original idth. Thus the width doubling operation applied on the pair ( W, X ) doubles the width of most ofthe grids on the vertices of the grids in X , but there is a small error.Our main tool for building a large family of pairwise compatible labelled graphs is the width doublingoperation. Before we use it, let us prove some key properties of it. Claim 3.6.
Let W be a labelled graph and X be a set of labelled grids of W with the same width w and height h . If there are g grids in X then the size of F ( W, X ) is at least g/ − O (log( g ) ) .Proof. By definition we have g − ≤ a ≤ g and for any 1 ≤ i ≤ z , we have a i − − ≤ a i ≤ a i − . Fromthis it follows that g/ i − ≤ a i ≤ g/ i and log ( g ) − ≤ z ≤ log ( g ). By Stirling’s approximationthere is a constant c such that for every integer N (1) c N √ N ≤ (cid:18) N ⌊ N/ ⌋ (cid:19) . Now we get a lower bound for |F ( W, X ) | using (1) and the lower bound for a i as follows |F ( W, X ) | = z Y i =0 (cid:18) a i / ⌊ a i ⌋ (cid:19) ≥ z Y i =0 c a i / p a i / ≥ log ( g ) − Y i =0 g/ (2 · i ) − log c − log g . From this it is an easy task to conclude that |F ( W, X ) | ≥ g/ − O (log g ) as claimed. (cid:3) Claim 3.7.
Let W be a labelled graph and X a set of g labelled grids of W with the same width w and height h . Every labelled graph in F ( W, X ) contains at least g/ − O (log g ) labelled grids of width w on the vertices of the grids in X .Proof. At the i -th merging operation (0 ≤ i ≤ z −
1) exactly ⌈ a i ⌉ labelled grids of width 2 w werecreated, and we did not change these grids after their creation. Thus the number of grids of width 2 w is z − X i =0 l a i m ≥ log g − X i =0 g i +1 − g/ − O (log g ) . (cid:3) Claim 3.8.
Let W be a labelled graph and X a set of g labelled grids of W with the same width w and height h . For every labelled graph W ′ ∈ F ( W, X ) , the number of vertices in W ′ that are containedin labelled grids of width w (“leftover grids”) and that are contained in the grids of X in the originallabelled graph W is at most O ( wh √ g ) .Proof. In a labelled graph of F ( W, X ) those labelled grids of width w that are on the vertices of thegrids in X are all formed in the following way. We had to chose an even number of grids of the samewidth and height for a multiple merging operation, but there was an odd number of these grids. Inthese cases the grid that was not used in the multiple merging operation had height 2 i h and width w for some (1 ≤ i ≤ z ). Hence the number of vertices in such a grid is hw i . At every multiple mergingoperation there was at most one such grid, thus the number of vertices in all of these grids is at most log ( g ) X i =1 wh i = whO ( √ g ) . (cid:3) .3. The lower bound for K = 1 . Now we are ready to prove Theorem 1.5. First we present theproof for ℓ = 1, then we proceed by induction. The rough structure of the proof can be seen in Figure4. We note that the ℓ = 1 case follows from the main result of [14], however on one hand we need adifferent construction for the induction and on the other hand, here the construction is much simplersince we only prove an asymptotic result. Proof of Theorem 1.5 for k=2.
Let F be the family that contains only the empty (labelled) graph W on n vertices. We think of W as a labelled graph that contains an isolated vertex v and n − X be a subset of the vertices of W of size n − W , X ) to get the family of labelled graphs F = F ( W , X ). By Claim 3.8 every labelled graph in F contains at most O ( √ n ) vertices that are inlabelled grids of width 1. Therefore adding at most O ( √ n ) new vertices every labelled graph in F canbe completed to a 2-labelled graph by adding only additional edges to widen the grids. Let us denotethe resulting family of 2-labelled graphs by F . By Claim 3.6 |F | ≥ n/ − O (log( n ) ) . We apply the Z-swapping construction to F . From each labelled graph in F we get 2 n/ − O (log n ) Hamiltonian paths by Claim 3.4 and Claim 3.7. Therefore altogether we get 2 n − O ((log n ) Hamiltonianpaths on n + O ( √ n ) vertices and the proof is complete. Λ3.4. The lower bound for k = 2 ℓ . To prove Theorem 1.5 for ℓ = 1 we used one width doublingoperation, for general ℓ we will use many, see Figure 4. To simplify the proof we introduce the completewidth doubling operation. Complete width doubling operation
Let W be a labelled graph . The complete width doublingoperation applied to W produces a family of pairwise isomorphic and pairwise compatible labelledgraphs C ( W ) as follows. We partition the grids of W according to their dimensions (width and height).Let the classes of the partition be P , P , . . . P m . Now let C := { W } and for each 1 ≤ i ≤ m let C i := [ W j ∈C i − F ( W j , P i )then C ( W ) := C m . We will also use the following claim which intuitively says that a width doubling operation does notproduce too many new different shapes of grids.
Claim 3.9.
Let W be a labelled graph and X a set of labelled grids in W of the same width and heightand | X | = g . In a labelled graph in F ( W, X ) the number of different shapes of grids on the vertices of X is at most O (log g ) .Proof. After every multiple merging operation the number of shapes in a labelled graph increases byat most two and we used O (log g ) multiple merging operations. (cid:3) Claim 3.10.
Suppose that we are applying the width doubling operation to a pair ( W, X ) where X consists of labelled grids of height α . Then in the resulting family F ( W, X ) of labelled graphs the heightof the grids on the vertices of X is at most α log n/α .Proof. The number of grids of height α is obviously at most n/α . Therefore during the width dou-bling operation we applied at most log ( n/α ) multiple merging operations. After a multiple mergingoperation the height of the resulting grids is at most twice the height of the original ones. (cid:3) Contains only W the edgeless graphon n vertices.One width doubling operation. F ∗ = F Many, pairwisecompatible2-labelled graphs. The Z -swappingconstruction. H n − o ( n ) O (log n ) width doubling operations. F Many, pairwisecompatible4-labelled graphs. The Z -swappingconstruction. H n − o ( n ) O (log n ) width doubling operations. . . . O (log K n ) width doubling operations. F K Many, pairwisecompatible2 ℓ -labelled graphs. The Z -swappingconstruction. H K n − o ( n ) ℓ +1-neighborseparatedHamiltonian paths Figure 4.
The induction steps. roof of Theorem 1.5 for every ℓ . Let W be the empty graph on n vertices and let us apply the complete width doubling operation ℓ -times to W . Let the resulting series of families of labelled graphs be F , F , . . . , F ℓ . We wish totransform the labelled graphs in F ℓ into 2 ℓ -labelled graphs. We show that this can be done by addingonly o ( n ) additional vertices and widening every labelled grid using these. Claim 3.11.
The number of vertices in a labelled graph in F ℓ which do not belong to a labelled gridof width ℓ is o ( n ) .Proof. By Claim 3.9 the number of different shapes of grids in F is at most O (log n ). Since a completewidth doubling operation is a width doubling operation on every different shape, in F i the number ofdifferent shapes of grids is at most O (log n i ). Every grid in F ℓ that has width less than 2 ℓ , must bea “leftover grid” in some of the many width doubling operations during the complete width doublingoperations that resulted in F ℓ therefore there is a single grid with this shape. Thus it is enough tobound the number of vertices in a single grid in F ℓ .The width of all the grids in F ℓ is bounded by a constant 2 ℓ , the height of the grids can be boundedby Claim 3.10 as follows.In F the maximal height of a grid is 1. If the maximal height of a grid in F j is α = O ( n − ε ) fora constant ε = ε ( j ) then the maximal height of a grid in F j +1 is at most α log ( n/α ) = α p n/α = α / n / = O ( n − ε/ ). Therefore setting ε (0) = 1 and ε ( j + 1) = ε ( j ) /
2, by induction we have that in F ℓ the height of the grids is at most n − ε for a fixed epsilon. Therefore the total number of verticesin grids with width smaller than 2 ℓ is at most O (2 ℓ n − ε log n ℓ ) = o ( n ) as claimed. (cid:3) By Claim 3.11 with the addition of at most o ( n ) new vertices we can build a family of 2 K -labelledgraphs from F ℓ , let us call this new family F final . Claim 3.12.
Applying the Z -swapping construction to F final we get n − o ( n ) Hamiltonian paths.Proof.
Let F be a family of labelled graphs which consists of pairwise isomorphic labelled graphs, let g ( F ) be the number of labelled grids in F . Let the value of such a family of labelled graph be thequantity v ( F ) := |F | g ( F ) . The value of the family that contains only the empty graph on n verticesis clearly 2 n . Suppose that we applied a width doubling operation to every graph in a family F witha set X of size g , then by Claim 3.6 |F | is increased by at least 2 g/ − O (log g ) . But by Claim 3.7 thequantity 2 g ( F ) is decreased by at most 2 g/ O (log g ) . Therefore the value of F is decreased by at most O (log g ) = O (log n ). Since F ℓ is the result of at most O (log n ℓ ) width doubling operations, the valueof F ℓ is at least 2 n − O (log n ℓ +2 ) = 2 n − o ( n ) . Since the number of Hamiltonian paths that we get afterapplying the Z -swapping construction to F final is exactly v ( F final ) the proof is complete. (cid:3) Since the number of vertices of the labelled graphs in F final is n + o ( n ), the proof of Theorem 1.5 iscomplete. 4. Connection with classical results
In this short section we discuss how determining P ( n, k ) and the compatibility of labelled graphsrelates to some classical theorems. Theorem 4.1 (B. Bollob´as ) . [1] Let H = { ( A , B ) , ( A , B ) , . . . , ( A m , B m ) } be a system of pairswhere for each i we have | A i | = a , | B i | = b . If A i ∩ B j = ∅ if and only if i = j then |H| ≤ (cid:18) a + ba (cid:19) . he system of pairs H satisfying the conditions of Theorem 4.1 is called cross-intersecting. Notethat the size of the ground set is not specified. Theorem 4.1 is sharp since we can take all possiblepartitions of [ a + b ]into two sets of size a and b . Theorem 4.1 has numerous variants, for our purposesthe following one is useful. Theorem 4.2 (Zs. Tuza ) . [22] Let H = { ( A , B ) , ( A , B ) , . . . , ( A m , B m ) } be a system of disjointpairs where for every i = j either A i ∩ B j = ∅ or A j ∩ B i = ∅ . If for every i , | A i | = a and | B i | = b then |H| ≤ ( a + b ) a + b a a b b . A system of pairs H satisfying the conditions of Theorem 4.2 is called weakly cross-intersecting.Theorem 4.2 is not known to be sharp, the current best lower bound is asymptotically 2 (cid:0) a + ba (cid:1) , althoughthe fractional relaxation of Theorem 4.2 is sharp, see [13].Suppose that we have a family F of labelled graphs. Let G be the family where we replace everylabelled graph with the pair ( A, B ) where A is the set of edges that get label a and B is the set ofedges that get label b . The family F contains pairwise compatible labelled graphs if and only if thefamily G consists of weakly cross-intersecting pairs.The original problem of determining P ( n, k ) can also be reformulated in terms of weakly cross-intersecting set systems. To each permutation π we associate a pair ( A π , B π ) as follows. Let A π contain the pairs of elements that are neighbors in π and B π contain the pairs of elements that areseparated by k − π . Two permutations are k -neighbor separated if and only if theirassociated pairs of sets are weakly cross intersecting. Observe that | A π | = n − | B π | = n − k + 1therefore using a = b = n in Theorem 4.2 we obtain that P ( n, k ) ≤ k for all k > Bounds for general fixed k In this section we improve the upper bound P ( n, k ) ≤ n provided by Theorem 4.2 for all k (seeSection 4). We conjecture that for every fixed k the order of magnitude of P ( n, k ) should be 2 n − o ( n ) .For even k we have seen that P ( n, k ) ≤ n − O ( log ( n )) . For odd k we will need the entropy function H ( x , x , x ) := P i =1 − x i log x i . We will use the factthat the entropy function is related to the asymptotic exponent of a multinomial coefficient. By astraightforward application of the the Stirling formula one can obtain the following well known result.
Claim 5.1.
For every positive x , x , x with the properties x + x + x = 1 there are positivepolynomials q , q so that for every ≤ n ∈ Z , (2) 1 q ( n ) 2 H ( x ,x ,x ) n ≤ (cid:18) nx n, x n, x n (cid:19) ≤ q ( n )2 H ( x ,x ,x ) n . Theorem 5.2.
For ≤ k fixed, we have P ( n, k ) ≤ H ( k − k , k − k , k ) n . Proof.
For every permutation π let us associate a coloring of the ground set with red , green and blue as follows. For every 1 ≤ i ≤ n let m = m ( i ) be such that m ≡ i mod 2 k −
2, and m ∈ [1 , k − π ( i ) is red if m ∈ { , , . . . , k − } green if m ∈ { k + 1 , k + 3 , . . . , k − } blue if m ∈ { , k } . Observe that two permutations which correspond to the same coloring cannot be k -neighbor sep-arated by the following reasoning. The colors of two neighboring elements in these permutations canbe: red-red, blue-red, green-green, green-blue. But the colors of two elements that are separated by − P ( n, k ) is at most the number of 3-coloringsof the ground set that we associated to the permutations. If n is divisible by 2 k then this number isexactly the multinomial coefficient (cid:18) n ( k − n k − , ( k − n k − , n k − (cid:19) ≤ q ( n )2 H ( k − k − , k − k − , k − ) n where q ( n ) is a fixed polynomial. If n is not divisible by 2 k then this number is at most anotherpolynomial factor away from the multinomial coefficient. All these polynomial factors can be safelyignored since a counterexample to the upper bound 2 H ( k − k , k − k , k ) n could be blown up to a coun-terexample with a larger exponent, contradicting the fact that we have an upper bound with only anadditional polynomial factor. (cid:3) Corollary 5.3. lim sup k →∞ P ( k ) ≤ Proof.
By Theorem 5.2 it is enough to check thatlim k →∞ H (cid:18) k − k − , k − k − , k − (cid:19) = 1 . (cid:3) Asymptotic lower bound.Theorem 5.4. lim inf k →∞ P ( k ) ≥ Proof.
Recall that for a fixed k and a fixed n if there are c n pairwise k -neighbor separated Hamil-tonian paths on n vertices then by a product construction we have the lower bound c ≤ P ( k ). Wewill construct such families for c arbitrarily close to 2 for every large enough k . Let r be a fixedinteger and k large so that r divides k and G a graph on k vertices that is the disjoint union of pathsof r vertices. Let F be the family of labelled graphs that can be formed from G by adding labels from { a, b } to every edge of G . Clearly |F | = 2 r − r k . For every W ∈ F we will construct a single Hamiltonian path H W so that two vertices that areconnected by an edge with label a in W are neighbors in H W and two vertices that are connected byan edge with label b in W are exactly k apart in H W . We do this in a way that the graphs H W useat most k + o ( k ) vertices. For each W ∈ F we wish to arrange the components of W and 6 kr + 9 r additional new vertices into a single ( k + 3 r ) × ( k + 3 r ) labelled grid. If we manage to do this, a suitableHamiltonian path can be obtained adding an isolated vertex and using the Z-swapping construction(we actually obtain two paths but this does not change the order of magnitude). Instead of arrangingthe labelled paths in W into a single labelled grid, we will fix a “grid-shape” of size ( k + 3 r ) × ( k + 3 r )and we aim to fill this completely using the paths of W and the additional isolated vertices (like apuzzle).Every labelled graph W ∈ F consists of vertex disjoint paths of length r − { a, b } . Observe that the number of possible labelling is at most 2 r − (actuallyless than this if r is at least 3 since the reversed version of a labelling can be considered the same)which is a constant. We say that paths of the same labelling are of the same type . As in other proofs,in a labelled grid we represent the edges that are labelled with a with horizontal edges and the onesthat are labelled with b with dashed vertical edges. Observe that using edges of the same type one cancompletely cover a diagonal strip in the large grid of width r , see part a ) of Figure 5. ) b) Figure 5. a) Shifted versions of paths of the same type fit together. b) We cover largepieces of the ( k + 3 r ) × ( k + 3 r ) grid using these strips. When we run out of a giventype of paths, we start using another type in the same strip. We will use a few isolatedvertices to fill in the gaps at the type change.We divide the ( k + 3 r ) × ( k + 3 r ) grid into stripes of width r and from top to the bottom, from leftto the right we fill those stripes with labelled paths of the same type. If we run out of a type, we startusing paths of another type in the same stripe as can be seen in part b ) of Figure 5. Claim 5.5.
For large enough k , every labelled path of W fits into the ( k + 3 r ) × ( k + 3 r ) grid.Proof. Suppose to the contrary that there are still paths from W , let us count the number of uncoveredvertices in the grid. there are two reasons why a vertex is uncovered: Either we got to the edge ofthe grid and the path that would cover our vertex sticks out of the grid, or we ran out of a type andstarted to use another. The number of vertices that are uncovered because the edge of the grid istoo close is at most 4 kr , since the distance of these vertices from the edge of the grid is at most r .The number of vertices that are uncovered because we ran out of a type and started to use anotheris even smaller. The number of type changes is at most 2 r − . At a single type change it is easyto see that the number of uncovered vertices is O ( r ). Therefore the number of vertices that areuncovered because of a type change is at most O (2 r r ). Thus the number of covered vertices is at least( k + 3 r ) − kr − O (2 r − r ) = k + 2 kr − O (2 r − r ). Since r is a constant, for large enough k thisquantity is larger than k , the total number of vertices in W which is a contradiction. (cid:3) Since every labelled path in W fits into the ( k + 3 r ) × ( k + 3 r ) grid, for every labelled graph W ∈ F (recall that labelled graphs in F have k vertices), we can construct a Hamiltonian path on ( k + 3 r ) vertices so that the new system of Hamiltonian paths is ( k + 3 r + 1)-neighbor separated. Therefore2 r − r k ≤ P (( k + 3 r ) , k + 3 r + 1)for any fixed r . For fixed r and k and n tending to infinity a product construction implies2 r − r n − o ( n ) ≤ P ( n, k + 3 r + 1) . Since we can choose r arbitrarily large, we have2 ≤ lim inf k →∞ P ( k )as claimed. Theorem 5.4 and Corollary 5.3 together implieslim k →∞ P ( k ) = 2 . The behaviour of P ( n, n )For our lower bound on P ( n, n ) we will use the following lemma. Lemma 6.1.
When n ≥ is even, there are three perfect matchings M , M , M on n vertices suchthat all three pairwise unions form a Hamiltonian cycle.Proof. Let M := { (1 , , (3 , , . . . , ( n − , n ) } M := { ( n, , (2 , , . . . , ( n − , n − } . Note that M ∪ M is a Hamiltonian cycle. If n = 4 k + 2 it is easy to see that the matching M := { (1 , n/ , (2 , n/ , . . . , ( n/ , n ) } completes both M and M to a Hamiltonian cycle. If n = 4 k then consider the following matchingthat consists of the shifts of the edges { (1 , , (3 , } : M ′ := { (1 , , (3 , , (5 , , (7 , , . . . (1 + 4 l, l ) , (3 + 4 l, l ) , . . . , ( n − , n ) , ( n − , } . ( M ∪ M ′ is a Hamiltonian cycle but M ∪ M ′ is the union of two cycles of length n/ M by changing only two edges in M ′ : M := ( M ′ \ { (1 , , (3 , } ) ∪ { (1 , , (4 , } . It is again easy to see that M completes both M and M to a Hamiltonian cycle and the proof iscomplete. (cid:3) Remark.
It is conjectured by several authors that for all n , the maximal number of perfect matchingson n vertices where every pairwise union is a Hamiltonian cycle is n −
1, see the survey [21]. If such aset of perfect matchings exists, it is called a perfect -factorization of the complete graph . Proof of Theorem 1.8.
We define a labelled H -cycle to be a labelled graph that is a Hamiltonian cyclewith a single edge that gets label b and every other edge gets label a . Observe that P ( n, n ) is themaximal number of pairwise compatible labelled H -cycles. For the upper bound:
Suppose that we have a family H of pairwise compatible labelled H -cycleson the vertex set [ n ]. We define a new graph G on the vertices [ n ]. Two vertices x and y in G areadjacent if there is a labelled H -cycle in H that contains the edge { x, y } with label b . Since everylabelled H -cycle contains only one edge of label b and H consists of pairwise compatible labelledgraphs we have E ( G ) = |H| . Claim 6.2. ∆( G ) ≤ Proof.
Suppose that we have a vertex in G that has degree at least 4. Choose four edges that areincident to this vertex. These edges correspond to pairwise compatible labelled H -cycles. The edgesthat get label b in these labelled H -cycles form a star. Therefore for each such H -cycle H , there isat most one other among the other three, that has its edge of label b contained in H with label a .Hence there are at most four compatible pairs among these four labelled H -cycles. But for pairwisecompatibility there must be at least six, a contradiction. (cid:3) ince ∆( G ) ≤
3, by the sum of the degrees in G , we have | E ( G ) | ≤ (cid:4) n (cid:5) and the proof of the upperbound is complete. For the lower bound: If n is even, by Lemma 6.1 there are three perfect matchings M , M , M on [ n ] such that every pairwise union of these is a Hamiltonian cycle. We define three sets of labelled H -cycles as follows (all the indices in the following definition are understood modulo 3). For i = 1 , , E i be a set of labelled H -cycles that contain every edge of M i and M i +1 . One edge of M i gets label b and every other edge gets label a . Let F = E ∪ E ∪ E . Clearly | E i | = n/ i thus |F | = n . It is easy to see that F consists of pairwise compatiblelabelled H -cycles. If n is odd, we proceed similarly as in the even case. We take the three perfectmatchings M , M and M on n − n −
1. We change the matchings byreplacing a single edge in M and M with an edge that is incident to the n -th vertex in such a waythat no two matchings share an isolated vertex. After this change we proceed in the same way as inthe even case. The only difference is that now we only have |F | = n − = (cid:4) n (cid:5) − . (cid:3) Remark.
It is not hard to show that when n is odd, there are no three graphs M , M , M with thefollowing properties • M , M , M are edge disjoint • Each pairwise union from { M , M , M } is a subgraph of a Hamiltonian cycle • | E ( M ) | + | E ( M ) | + | E ( M | = (cid:4) n (cid:5) . Therefore the strategy that we used to show that for even n , n ≤ P ( n, n ) can not be used when n isodd. 7. Open problems and concluding remarks
In [14] the authors determined the exact value of P ( n,
3) and its rough order of magnitude is 2 n − o ( n ) .The main results of the present paper suggest that the order of magnitude of P ( n, k ) is also 2 n − o ( n ) when k is odd.In the present paper the methods that lead to the exact value of P ( n,
3) were applied to the caseof P ( n, ℓ + 1) and in these special cases we managed to prove Conjecture 1.4. Our results in Section5 imply that (2 − ε ( k )) n ≤ P ( n, k ) ≤ (2 + ε ( k )) n where ε ( k ) tends to zero with k tending to infinity.Therefore Conjecture 1.4 holds asymptotically.When proving Conjecture 1.4 for k = 2 ℓ + 1, we constructed pairwise compatible labelled graphswith a linear number of labelled grids that got us 2 n − o ( n ) permutations. It would be useful to constructa family of 2 n − o ( n ) pairwise compatible labelled graphs of any fixed width k with a single labelled grid.If we can do this, it not only implies P ( n, k + 1) ≈ n − o ( n ) , but also P ( n, kℓ + 1) ≈ n − o ( n ) forevery positive integer ℓ , as follows. We can divide n vertices into ℓ equal parts and take a single-gridconstruction on each part and take the “product” of these constructions. This way our labelled graphscontain exactly ℓ labelled grids and we can merge these as the W part of Figure 3 to get a single gridof width kℓ . This way we got a set of 2 n − o ( n ) compatible labelled graphs of width kℓ . Note that justmerging the grids as the W part of Figure 3 decreases the exponent if we have a linear number oflabelled graphs.Observe that two labelled grids are compatible if and only if their “transposed” versions are com-patible. The transposed version of a grid with width 2 and height n/ n/ n/ P ( n, n/ onsider the middle edge of each permutation and take only those that have the most frequent middleedge, this way we only lose a factor of (cid:0) n (cid:1) ). Therefore for any fixed k we havelim n →∞ n p P ( n, n/ ≤ lim n →∞ n p P ( n, k + 1) . Since P ( n, ≤ n , we have lim n →∞ n p P ( n, n/ ≤ Question 7.1.
What is the limit of n p P ( n, n/ when n tends to infinity? Definition 7.2.
The graph G is the union of two graphs G , G on the same vertex set if V ( G ) = V ( G ) = V ( G ) and E ( G ) = E ( G ) ∪ E ( G ) . We say that two Hamiltonian paths H , H on thesame vertex set [ n ] are G -different if G is a (not necessarily induced) subgraph of H ∪ H . Let H ( n, k ) be the maximal number of C k -different Hamiltonian paths on n vertices. Observe that H ( n,
3) = P ( n, H ( n, k ) for fixed k . For odd k , our best lower boundsfor H ( n, k ) come from the inequality P ( n, k ) ≤ H ( n, k ) and the results of the present paper. For odd k , the upper bound H ( n, k ) ≤ n − o ( n ) holds by the argument used in Proposition 1.1.For even k , H ( n, k ) is much larger than P ( n, k ). In [8] it is proven that n n − o ( n ) ≤ H ( n, ≤ n n − o ( n ) . The lower bound is easy: The family of directed Hamiltonian paths of [ n ] where for every i ,the (2 i + 1)-th vertex of every Hamiltonian path is the vertex (2 i + 1) ∈ [ n ] satisfies the conditions. It isnot hard to generalize this construction to yield n k n − o ( n ) ≤ H ( n, k ). The upper bound in [8] is muchmore involved, and in a subsequent paper it will be generalized to yield H ( n, k ) ≤ n (cid:16) − ck (cid:17) n − o ( n ) .Therefore our knowledge of H ( n, k ) can be summarized as follows n k n − o ( n ) ≤ H ( n, k ) ≤ n (cid:16) − ck (cid:17) n − o ( n ) . For P ( n, k ) and H ( n, k + 1), if k is getting larger, our bounds are improving, but in the case of H ( n, k ) they are getting worse. 8. Acknowledgement
We would like to thank G´abor Simonyi and G´eza T´oth for their help which improved the qualityof this manuscript.
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E-mail address , Istv´an Kov´acs: [email protected]
Alfr´ed R´enyi Institute of Mathematics, Hungarian Academy of Sciences
E-mail address , Daniel Solt´esz: [email protected]@math.bme.hu