aa r X i v : . [ m a t h . C O ] M a y FAMILIES OF LOCALLY SEPARATEDHAMILTON PATHS
J´anos K¨orner e–mail: [email protected]
Angelo Monti e–mail:
Sapienza University of RomeITALY
Abstract
We improve by an exponential factor the lower bound of K¨orner and Muzi for thecardinality of the largest family of Hamilton paths in a complete graph of n verticesin which the union of any two paths has degree 4. The improvement is through anexplicit construction while the previous bound was obtained by a greedy algorithm.We solve a similar problem for permutations up to an exponential factor. Keywords
Hamilton paths, graph-difference, permutations, intersection prob-lems
AMS Subject classification numbers Introduction
Greedy algorithms play a central role in combinatorial problems related to informationtheory. The classical example for this is the code distance problem asking for the max-imum cardinality of a set of binary sequences of length n any two of which differ in atleast αn coordinates where α is a fixed constant. Gilbert [4] and Varshamov [12] showedindependently that for α ∈ [0 , /
2) the maximum cardinality is at least exp n (1 − h ( α ))where h is the binary entropy function h ( t ) = − t log − (1 − t ) log (1 − t ) . This result,called the Gilbert–Varshamov bound, is proved by a greedy algorithm. Considerable ef-fort went into trying to strengthen this bound, and although it was recently improved bya linear factor [6], [13], it is believed by many to be exponentially tight.In an attempt to generalise Shannon’s zero–error capacity problem [10] from fixed–length sequences of elements of a finite alphabet to permutations and then to subgraphsof a fixed complete graph, cf. [7], [2] and [8], one of the main construction techniques isagain based on greedy algorithms. It is crucial to understand the true role of greedy algo-rithms in our context. We will study two different combinatorial problems of information–theoretic flavour and discuss the constructions obtained by the greedy algorithm in bothcases. For the general framework and especially for Shannon’s zero–error capacity problemwe refer the reader to the monograph [3].
The first problem we are to deal with is about Hamilton paths in the fixed complete graph K n with vertex set [ n ] = { , , . . . , n } . We will say that two such paths are crossing iftheir union has degree 4. Let us denote by Q ( n ) the maximum cardinality of a family ofHamilton paths from K n in which any pair of paths is crossing. It was proved in [8] (inTheorem 3) that ( n − ⌊ n/ ⌋ !(1 + √ n ≤ Q ( n ) ≤ n ! ⌊ n/ ⌋ !2 ⌊ n/ ⌋ . The lower bound was achieved by a greedy algorithm. The core of this proof was anupper bound on the number of Hamilton paths not crossing a fixed path. Here we give anexplicit construction resulting in a larger lower bound. The main message of this result isthat for the present problem the greedy algorithm is not tight. The improvement is in thesecond term of the asymptotics of Q ( n ) and it could not be otherwise for the leading termsin the lower and upper bounds for Q ( n ) in [8] coincide. In fact, both of the bounds in theabove theorem are of the form 2 cn ( dn )! where c and d are constants. The leading term isthe factorial and the constant d in it is the same in both of the bounds. The differenceis in the constant in the exponential factor. Our new result increases the constant in theexponential factor in the lower bound in Theorem 3 in [8].More precisely, we have 1 heorem 1 ( ⌊ n/ ⌋ − ⌊ n/ ⌋ ≤ Q ( n ) . Before proving our result we observe that( n − ⌊ n/ ⌋ !(1 + √ n < (cid:16)
21 + √ (cid:17) n · ⌈ n/ ⌉ ! ≤ ( ⌊ n/ ⌋ − ⌊ n/ ⌋ where the first inequality follows since( n − ⌊ n/ ⌋ ! ≤ n ! ⌊ n/ ⌋ ! = (cid:18) n ⌊ n/ ⌋ (cid:19) ( ⌈ n/ ⌉ )! < n ( ⌈ n/ ⌉ )!while the last inequality is obvious as √ < − / . Thus the new lower bound of thistheorem improves on the one in [8] quoted above.
Proof.
We consider the complete bipartite graph K ⌊ n/ ⌋ , ⌈ n/ ⌉ . For the ease of notation let ussuppose that this graph is defined by the bipartition of [ n ] into the sets A and B where A = [ ⌊ n/ ⌋ ] is the set of the first ⌊ n/ ⌋ natural numbers while B = [ n ] − A. Let us fix apermutation of the elements of B. Then any permutation of A defines a Hamilton pathin K ⌊ n/ ⌋ , ⌈ n/ ⌉ where the first element of the path is the first element of the permutationof B . The path alternates the elements of A and B in such a way that the elements of B are in the odd positions along the path and in the order defined by the permutation of B we have fixed. Likewise, the relative order of the elements of A following the path isthe one given by the permutation of A. Every Hamilton path in this family can thus beidentified with some permutation of the set A. We represent each permutation as a linearorder of A. Notice now that if there is an element a ∈ A such that in two different linear ordersits position differs by at least 2 while none of these positions is the last one, then thevertex a has disjoint pairs of adjacent vertices in the two Hamilton paths. In other words,the vertex a has degree 4 in the union of these paths. We will say that such a pair ofpermutations is 2-different. We will now construct a sufficiently large family of pairwise2–different permutations. To this end, we first observe that if two permutations are 2–different, then in their respective inverses there is some natural number whose imagesdiffer by at least two. K¨orner, Simonyi and Sinaimeri have shown (cf. Theorem 1 in [9])that for any natural number m the set [ m ] has a family of exactly m !2 ⌊ m/ ⌋ permutationssuch that for any two of them there is at least one element of [ m ] that has images differingby at least two. Hence the inverses of these permutations form a family satisfying ouroriginal condition, except for the fact that the a whose positions in a pair of permutationsdiffer by at most two positions might be in the last position in one of them. To excludethis event, we choose a subfamily of our permutations in which the last element in all thepermutations is the same. Choosing m = | A | we complete the proof.2 The construction in the last proof seems sufficiently natural for us to believe thatit might be asymptotically optimal at least for Hamilton paths in a balanced completebipartite graph. In other words, let B ( n ) denote the largest cardinality of a family ofHamilton paths in K ⌊ n/ ⌋ , ⌈ n/ ⌉ any two members of which are crossing. If the answer ispositive, one might go even further and ask whether the lower bound in Theorem 1 isasymptotically tight. In [8] where the previous problem was introduced, it was presented alongside with a fairlygeneral framework for extremal combinatorics with an information-theoretic flavour. Thisframework is in terms of two (mostly but not necessarily disjoint) families of graphs on thesame vertex set [ n ] = { , , . . . , n } . If we denote these families by F and D , respectively,then M ( F , D , n ) stands for the largest cardinality of a subfamily G ⊆ F with the propertythat the union of any two of its different members belongs to D . Here the union of twographs on the same vertex set is the graph whose edge set is the union of those of the twographs. This framework is the fruit of an attempt to describe a consistent part of extremalcombinatorics where information theoretic methods seem to be relevant. In [8] it wasemphasised that when the families F and D are disjoint, the question about M ( F , D , n )is fairly information–theoretic in nature and there never is a natural candidate for anoptimal construction; much as it happens for the graph capacity problem of Shannon’s.As a matter of fact, in each of these cases one is interested in the largest cardinality ofa family of mathematical objects (graphs or strings) any pair of which are different insome specific way. Intuitively, the objects in these families are not only different, but alsodistinguishable in some well–defined sense inherent to the problem. In other words, theoptimal constructions are, in a vague and broad sense, similar to the channel codes ininformation theory.If, however, F = D , the solution of our problems seems to be of a completely differ-ent nature, and, in particular, any analogy with codes disappears. Rather, the optimalsolutions often are so–called kernel structures, and even when this is not the case, kernelstructures give good constructions for the problem. Kernel structures are families of ob-jects having in common a fixed projection, called the kernel. This means that in a vagueand broad sense, these objects are similar. We are trying to build up a dichotomy distin-guishing those problems for which kernel structures can be defined (and often are evenoptimal) and those where the optimal structures are like ”codes” in information theory.It would be highly interesting to get a better understanding of this dichotomy.At the surface, the dichotomy for the problems of determining M ( F , D , n ) is basedon whether or not F = D . However, this view is too simplistic. As a matter of fact, ournext problem will illustrate the lack of an easy way to distinguish between the two cases.More precisely, we will give an example where the families F and D are disjoint, and the3olution of the problem still has a kernel structure. To explain our example, let F cy bethe family of Hamilton cycles in the complete graph K n . However, let D be the family ofgraphs having at least one vertex of degree 3. The relation of having a vertex of degree3 in the union of two graphs is clearly irreflexive when restricted to cycles. In fact, theproblem of determining M ( F cy , D , n ) is, on the surface, no different from the one whichis the subject of Theorem 1. However, this impression is wrong and the present problemis completely different from the previous one. To understand why, it suffices to realisethat the union of two Hamilton cycles in K n has a vertex of degree 3 if and only if theyshare a common edge. This makes it possible to invoke kernel structures. Moreover, notsurprisingly, a quasi–optimal solution for this problem has a kernel structure. In fact, wehave Proposition 1 M ( F cy , D , n ) ≥ ( n − and this bound is tight for odd n. For even n we have M ( F cy , D , n ) ≤ ( n − n − Proof.
To prove the lower bound, consider the family of all Hamilton cycles in K n containinga fixed edge. To establish the upper bound, let first n be odd. By a classical constructionof Walecki (1890), cited in [1], for every odd n , the family of the Hamilton cycles of K n can be partitioned into ( n − K n into n − Hamilton cycles). Thisgives our upper bound for odd n . For even n we can only have families of n − pairwisedisjoint Hamilton cycles which gives the weaker bound for this case. ✷ A permutation of [ n ] is nothing but a (consecutively oriented) Hamilton path in thesymmetrically complete directed graph K n . (With a slight abuse of notation we denoteby K n also the directed graph in which every pair of distinct vertices is connected by twoedges going in opposite direction.) In the previous part of the paper we were dealing withproblems about Hamilton paths in (the undirected complete graph) K n such that for anypair of them, for at least one vertex in K n the union of its 1-neighbourhoods in the twopaths satisfied some disjointness condition. We now turn to an analogous problem aboutdirected paths.In a directed graph, the (open) 2–out–neighbourhood of a vertex a ∈ [ n ] consists ofthe vertices reachable from a by a (consecutively oriented) path of at most two edges.We will say that two directed Hamilton paths in K n are two–separated if for at least onevertex of K n the union of its 2–out–neighbourhoods in the two paths contains exactly4 vertices. Formulating this condition in terms of permutations means that for some a ∈ [ n ] its two immediate successors in the two linear orders defining the two respectivepermutations are axy and avw where x, y, v, w are four different elements of [ n ] . Thisrelation is seemingly very similar to the one underlying Theorem 1. As a matter offact, in the permutation language, to two crossing Hamilton paths there correspond twopermutations such that for some a ∈ [ n ] its neighbours are completely different, meaningthat we have xay and vaw in the two permutations and x, y, v, w are all different. If allpairs of paths in a Hamilton path family are crossing then for any one of the paths onlyone of the corresponding two permutations can be in the family, and it can be eitherone of the two that correspond to the same path. Somewhat surprisingly, the cardinalityof the largest family of pairwise two–separated Hamilton paths is of a different order ofmagnitude from the one in Theorem 1. In fact, we have the following almost tight result. Theorem 2
Let R ( n ) stand for the maximum cardinality of a family of pairwise two–separated Hamilton paths in K n . Then n ! n n − ≤ R ( n ) ≤ n !2 ⌊ n/ ⌋ . Proof.
In this proof we will represent our Hamilton paths as permutations. We will say thattwo permutations are two–separated if they correspond to two–separated Hamilton paths.We associate with any permutation a sequence of unordered couples of elements of [ n ] . The permutation π = π . . . π n is associated with the sequence of unordered and disjointcouples { π , π } . . . { π i − , π i } . . . { π ⌊ n/ ⌋− , π ⌊ n/ ⌋ } . We will call this sequence of couplesthe couple order of π. We then partition the set of all permutations of [ n ] according totheir couple order. In each class the permutations it contains have the same couple orderand all the permutations with the same couple order belong to this class. We claim thatno two permutations in the same class are two–separated. Considering that the numberof classes is n !2 ⌊ n/ ⌋ , we will thus obtain our upper bound.In fact, let the permutations π ′ and π ′′ have the same couple order and let xy and vw be the two immediate successors of a in π ′ and π ′′ , respectively. We will distinguishtwo cases. In the first case a and x appear in the same couple { a, x } of the couple order,while in the second case a is in a different couple from x and y , which then have toappear together in the subsequent couple. In the first case either a and x appear in thesame order ( a, x ) in both permutations and we are done, or their order is different, butthen y must appear in one of the two positions subsequent to a in π ′′ , contradicting ourhypothesis that π ′ and π ′′ are two–separated. It remains to see what happens if in thecouple order of π ′ the number a is in the couple preceding { x, y } . This means that in thecouple order a is in the same couple with some z ∈ [ n ] different from both x and y. Sinceour two permutations have the same couple order, we conclude that at least one of x and y is in one of the two positions immediately following a in π ′′ . R ( n ). To this end we will use a greedy algorithmto exhibit a large enough family of permutations with the required property. At each stepin the algorithm we choose an arbitrary permutation and eliminate from the choice spaceall those incompatible with the chosen one. This procedure goes on until the choice spacebecomes empty.To analyze this algorithm, we need an upper bound on the number of the permutationsincompatible with a fixed permutation ι of [ n ]. Without loss of generality we assume that ι is the identical permutation mapping each number in itself. Let π be a permutationincompatible with ι . We have thatfor each j , 1 ≤ j ≤ n −
2, if π j
6∈ { n, n − } then { π j +1 π j +2 } ∩ { π j + 1 , π j + 2 } 6 = ∅ (1)otherwise the two immediate successors of π j in the two permutations ι and π are fourdifferent elements of [ n ] , contradicting that ι and π are incompatible.If π j +1 ∈ { π j + 1 , π j + 2 } or respectively π j +2 ∈ { π j + 1 , π j + 2 } then we say that π j +1 orrespectively π j +2 is a close enough follower of π j . In this terminology property (1) meansthat at least one of π j +1 and π j +2 must be a close enough follower of π j .In the following we call run of big jumps in π any maximal (not extendable) sequence π j π j +1 . . . π j + t , 1 ≤ j ≤ n − t such that π j +1 − π j
6∈ {− , − , , , } and π k +1 − π k , , } , j < k < j + t . We will refer to the position j as the head of the run and to thepositions k of the run with k ≥ j + 2 as tail positions of the run. We call free positions of π the first position of the entire permutation, the second position in any run and thetwo positions immediately following those of either n or n −
1. All the other positions willbe called constrained. (We will soon explain the meaning of this terminology.) For anyconstrained position j not belonging to any run we have | π j − π j − | ≤ , implying that π j ∈ { π j − − , π j − − , π j − + 1 , π j − + 2 , π j − + 3 } and the same is true if j is the head of a run since in this case either π j − π j − > π j − π j − < − j in the tail of a run. By definition, position j − π j − − π j − ≥ π j − − π j − ≤ − . This implies that π j −
6∈ { π j − + 1 , π j − + 2 } and from property (1) we obtain π j ∈ { π j − + 1 , π j − + 2 } .We have just proved that for any constrained position j of π the value π j is close to atleast one of the values π j − and π j − .Let us now consider the permutations incompatible with ι having exactly r runs ofbig jumps. In these permutations there are at most r + 5 free positions and at most (cid:0) n − r +2 (cid:1) possibilities for these free positions (remember that, by definition, the first position isalways a free position and that the two free positions depending on n are adjacent to itand the same is true for the two free positions depending on n − (cid:18) n − r + 2 (cid:19) n r +5 n − ( r +5) . (2)6e will prove that in any incompatible permutation there can be at most three runs of bigjumps. This in turn implies that any incompatible permutation can have at most eightfree positions and from bound (2) we have that the number of permutation incompatiblewith permutation ι is bounded above by4 (cid:18) n − (cid:19) n n − . This means that the greedy algorithm will eliminate at most n n − permutations ateach step, yielding a family with the desired property and containing at least n ! n n − permutations, as claimed for the lower bound.To conclude the proof it remains to show that in a permutation π incompatible with ι there are at most three runs of big jumps. To this aim we now prove that any run of bigjumps of π that does not contain neither n nor n − π . The proof is bycontradiction. Let us assume that in π there is a run that does not contain neither n nor n − t be the last position of such a run.We distinguish two cases. In the first case, we suppose π t − > π t . By assumption, π t is the last position in the run of big jumps. Hence the last inequalitycan be satisfied only if π t ≤ π t − − . Thus π t is not close enough to π t − whence, by property (1), we must have π t +1 ∈{ π t − + 1 , π t − + 2 } . (Note that, by our assumption, π − t is not the last element in π ).But then π t +1 − π t ≥ , contradicting the assumption that π t is in the last position of the run of big jumps.In the second case we suppose π t − < π t . This implies π t ≥ π t − + 3 . (3)Thus π t is not a close enough follower of π t − so that once again, by property (1), we musthave π t +1 ∈ { π t − + 1 , π t − + 2 } . In particular, we see that π t +1 < π t . However, since π t +1 is not part of the run of big jumps, the only possibility left is π t +1 = π t − . (4)7oreover since π t is not a close enough follower of π t − this role must be taken by π t +1 .Hence from (4) and from (3) it follows that π t = π t − + 3 (5)Thus, by (4) and (5), we have π t +1 = π t − + 2 . (6)Let us see what we can say at the light of this about π t − . Note that π t − π t − = 3which implies that position t − t − π t − − π t −
6∈ {− , , } . Hence we must have π t − ≤ π t − − π t − ≥ π t − + 2. Moreover, property (1) implies { π t − + 1 , π t − + 2 } ∩ { π t − , π t } 6 = ∅ , whence the only choice left is π t − = π t − + 2 (remember that π t = π t − + 3), and this isimpossible by equality (6). ✷ We have already mentioned the need for a meta–conjecture to describe for which problemswithin our framework does the optimal solution have a kernel structure. It seems to usthat this can happen only in case of one–family problems and it will never be the case ifthe family D in a two–family problem is not monotone. (A class F of graphs is monotone if G ∈ F and G ⊆ H imply H ∈ F . Note, in particular, that the family D of graphs havinga vertex of degree 3 is not monotone.) Our problems belong to a class of combinatorialproblems introduced in [11] where they are referred to as intersection problems. In theeyes of the authors of [11] an intersection problem is in terms of the maximum cardinalityof a family of objects such that the pairwise intersections of these objects have someprescribed property. (Even if, in this paper, we are considering pairwise unions, we mighthave considered, equivalently, the intersections of the complementary objects.) We feelthat it is more appropriate to reserve the term intersection problem to the case F = D when in fact we end up with an intersection problem in disguise, in the sense of [11]. Inour opinion, an intersection problem is a one–family problem. In such problems one asksfor the largest cardinality of a family of objects, typically subgraphs of K n with a givenproperty such that the intersection of any two members of the family is still in the family.One often considers as intersection problems only those about the maximum cardinalityof a family of graphs such that the intersection of any two of them is non–empty.A more delicate question is to understand when exactly can the optimal solution ofa problem of our kind be obtained through a greedy algorithm. Our two main resultscan hopefully contribute to a better understanding of this issue. At a first glance one istempted to believe that for an information–theoretic (genuinely two–family) problem thegreedy algorithm often gives close-to-optimal constructions. Our question is not preciseenough and we do not expect a precise answer at this stage.8 Acknowledgement
We are grateful to Miki Simonovits for stimulating discussions about the true nature ofintersection problems. 9 eferences [1] J. Akiyama, M. Kobayashi, G. Nakamura, Symmetric Hamilton cycle decompositionsof the complete graph.
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