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1.
It is shown that if a chain complete ordered set does not have k+1 pairwise disjoint maximal chains for some finite k, then the minimum size of a cutset is equal to the maximum size of a collection of pairwise disjoint maximal chains. This answers a question of Pouzet and Zaguia.The author was supported in part by ONR grant N00014-85K-0494 and NSERC grants 69-3378, 69-0259, and 69-1325. 相似文献
2.
We prove the following results which are related to Menger's theorem for (infinite) ordered sets. (i) If the space of maximal chains of an ordered set is compact, then the maximum number of pairwise disjoint maximal chains is finite and is equal to the minimum size of a cutset, (i.e. a set which meets all maximal chains). (ii) If the maximal chains pairwise intersect, then the intersection of finitely many is never empty. One corollary of (ii) is that, if the maximal chains pairwise intersect and if one of the maximal chains is complete, then there is an element common to all maximal chains. 相似文献
3.
Let P be a chain complete ordered set. By considering subsets which meet all maximal chains, we describe conditions which imply that the space of maximal chains of P is compact. The symbols P
1 and P
2 refer to two particular ordered sets considered below. It is shown that the space of maximal chains ( P) is compact if P satisfies any of the following conditions: (i) P contains no copy of P
1 or its dual and all antichains in P are finite. (ii) P contains no proper N and every element of P belongs to a finite maximal antichain of P. (iii) P contains no copy of P
1 or P
2 and for every x in P there is a finite subset of P which is coinitial above x. We also describe an example of an ordered set which is complete and densely ordered and in which no antichain meets every maximal chain. 相似文献
4.
and let be the collection of all subsets of [ n] ordered by inclusion. is a cutset if it meets every maximal chain in , and the width of is the minimum number of chains in a chain decomposition of . Fix . What is the smallest value of such that there exists a cutset that consists only of subsets of sizes between m and l, and such that it contains exactly k subsets of size i for each ? The answer, which we denote by , gives a lower estimate for the width of a cutset between levels m and l in . After using the Kruskal–Katona Theorem to give a general characterization of cutsets in terms of the number and sizes of
their elements, we find lower and upper bounds (as well as some exact values) for .
Received September 4, 1997 相似文献
5.
Fix integers n and k with n≥ k≥3. Duffus and Sands proved that if P is a finite poset and n≤| C|≤ n+( n− k)/( k−2) for every maximal chain in P, then P must contain k pairwise disjoint maximal antichains. They also constructed a family of examples to show that these inequalities are tight. These examples are two-dimensional which suggests that the dual statement may also hold. In this paper, we show that this is correct. Specifically, we show that if P is a finite poset and n≤| A|≤ n+( n− k)/( k−2) for every maximal antichain in P, then P has k pairwise disjoint maximal chains. Our argument actually proves a somewhat stronger result, and we are able to show that an analogous result holds for antichains. 相似文献
6.
For each integer k≥3, we find all maximal intervals Ik of natural numbers with the following property: whenever the number of elements in every maximal chain in a finite partially ordered set P lies in Ik, then P contains k pairwise disjoint maximal antichains. All such Ik are of the form
7.
We say that an ordered set P is spanned by a family C of chains if P=( P, ) is the transitive closure of {( C, | C) C C. It is shown that there is a function h: such that if P is spanned by k< chains, then P has a finite cutset-number h(k) (i.e. for any xP, there is a finite set F of size | F| h(k)–1, such that the elements of F are incomparable with x and { x} F meets every maximal chain of P). The function h is exponentially bounded but eventually dominates any polynomial function, even if it is only required that there are at most h(k) pairwise disjoint maximal chains in P, whenever P is spanned by k< chains. 相似文献
8.
Let F be a partially ordered set (poset). A poset P is called F-free if P contains no subposet isomorphic to F. A finite poset F is said to have the maximal element property if every maximal F-free subposet of any finite poset P contains a maximal element of P. It is shown that a poset F with at least two elements has the maximal element property if and only if F is an antichain or F ≅ 2 + 2. 相似文献
9.
AbstractGiven a topological space X = ( X, T ), we show in the Zermelo-Fraenkel set theory ZF that: Every locally finite family of open sets of X is finite iff every pairwise disjoint, locally finite family of open sets is finite. Every locally finite family of subsets of X is finite iff every pairwise disjoint, locally finite family of subsets of X is finite iff every locally finite family of closed subsets of X is finite. The statement “every locally finite family of closed sets of X is finite” implies the proposition “every locally finite family of open sets of X is finite”. The converse holds true in case X is T4 and the countable axiom of choice holds true. We also show:It is relatively consistent with ZF the existence of a non countably compact T1 space such that every pairwise disjoint locally finite family of closed subsets is finite but some locally finite family of subsets is infinite. It is relatively consistent with ZF the existence of a countably compact T4 space including an infinite pairwise disjoint locally finite family of open (resp. closed) sets. 相似文献
10.
For a partially ordered set P and an element x of P, a subset S of P is called a cutset for x in P if every element of S is noncomparable to x and every maximal chain of P meets { x}∪ S. We let c( P) denote the smallest integer k such that every element x of P has a cutset S with ‖ S‖? k: If c( P)? n we say that P has the n-cutset property. Our results bear on the following question: given P, what is the smallest n such that P can be embedded in a partially ordered set having the n-cutset property? As usual, 2 n denotes the Boolean lattice of all subsets of an n-element set, and B n denotes the set of atoms and co-atoms of 2 n . We establish the following results: (i) a characterization, by means of forbidden configurations, of which P can be embedded in a partially ordered set having the 1-cutset property; (ii) if P contains a copy of 2 n , then c( P)?2 [n/2]?1; (iii) for every n>3 there is a partially ordered set P containing 2 n such that c( P)< c(2 n ); (iv) for every positive integer n there is a positive integer N such that, if B m is contained in a partially ordered set having the n-cutset property, then m? N. 相似文献
11.
A short proof of the following theorem is given: Let P be a finite partially ordered set. If the maximal number of elements in an independent subset of P is k, then P is the union of k chains. 相似文献
12.
A result of G. Chartrand, A. Kaugars, and D. R. Lick [Proc Amer Math Soc 32 (1972), 63–68] says that every finite, k‐connected graph G of minimum degree at least ?3 k/2? contains a vertex x such that G? x is still k‐connected. We generalize this result by proving that every finite, k‐connected graph G of minimum degree at least ?3 k/2?+ m?1 for a positive integer m contains a path P of length m?1 such that G? V( P) is still k‐connected. This has been conjectured in a weaker form by S. Fujita and K. Kawarabayashi [J Combin Theory Ser B 98 (2008), 805–811]. © 2009 Wiley Periodicals, Inc. J Graph Theory 65: 61–69, 2010. 相似文献
13.
Consider the scalar kth order linear difference equation: x( n + k) + pi( n) x( n + k - 1) + … + pk( n) x( n) = 0 where the limits qi=lim n→∞ Pi( n) ( i=1,…, k) are finite. In this paper, we confirm the conjecture formulated recently by Elaydi. Namely, every nonzero solution x of (?) satisfies the same asymptotic relation as the fundamental solutions described earlier by Perron, ie., ?= lim sup n→∞ | x( n)| is equal to the modulus of one of the roots of the characteristics equation χ k + q 1χ k?1+…+ qk=0. This result is a consequence of a more general theorem concerning the Poincaré difference system x( n+1)=[ A+ B( n] x( n), where A and B( n) ( n=0,1,…) are square matrices such that ‖ B( n)‖ →0 as n → ∞. As another corollary, we obtain a new limit relation for the solutions of (?). 相似文献
14.
A regression is a function g from a partially ordered set to itself such that g( x)≦ x for all z. A monotone k-chain is a chain of k elements x
1< x
2 <...< x
k
such that g( x
1)≦ g( x
2)≦...≦ g( x
k
). If a partial order has sufficiently many elements compared to the size of its largest antichain, every regression on it
will have a monotone ( k + 1)-chain. Fixing w, let f( w, k) be the smallest number such that every regression on every partial order with size least f( w, k) but no antichain larger than w has a monotone ( k + 1)-chain. We show that f( w, k)=( w+1)
k
.
Dedicated to Paul Erdős on his seventieth birthday
Research supported in part by the National Science Foundation under ISP-80-11451. 相似文献
15.
For every positive integer k, there is a positive integer f(k) such that every finite digraph of minimum outdegree f(k) contains vertices x, y joined by k openly disjoint paths. 相似文献
16.
Let k be a fixed integer at least 3. It is proved that every graph of order (2 k ? 1 ? 1/ k) n + O(1) contains n vertex disjoint induced subgraphs of order k such that these subgraphs are equivalent to each other and they are equivalent to one of four graphs: a clique, an independent set, a star, or the complement of a star. In particular, by substituting 3 for k, it is proved that every graph of order 14 n/3 + O(1) contains n vertex disjoint induced subgraphs of order 3 such that they are equivalent to each other. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 159–166, 2007 相似文献
17.
A graph G is k-linked if G has at least 2 k vertices, and for any 2 k vertices x
1, x
2, …, x
k
, y
1, y
2, …, y
k
, G contains k pairwise disjoint paths P
1, …, P
k
such that P
i
joins x
i
and y
i
for i = 1,2, …, k. We say that G is parity-k-linked if G is k-linked and, in addition, the paths P
1, …, P
k can be chosen such that the parities of their length are prescribed. Thomassen [22] was the first to prove the existence
of a function f( k) such that every f( k)-connected graph is parity-k-linked if the deletion of any 4 k-3 vertices leaves a nonbipartite graph.
In this paper, we will show that the above statement is still valid for 50 k-connected graphs. This is the first result that connectivity which is a linear function of k guarantees the Erdős-Pósa type result for parity- k-linked graphs.
Research partly supported by the Japan Society for the Promotion of Science for Young Scientists, by Japan Society for the
Promotion of Science, Grant-in-Aid for Scientific Research and by Inoue Research Award for Young Scientists. 相似文献
18.
We give a sufficient condition for a simple graph G to have k pairwise edge‐disjoint cycles, each of which contains a prescribed set W of vertices. The condition is that the induced subgraph G[ W] be 2 k‐connected, and that for any two vertices at distance two in G[ W], at least one of the two has degree at least | V( G)|/2 + 2(k ? 1) in G. This is a common generalization of special cases previously obtained by Bollobás/Brightwell (where k = 1) and Li (where W = V( G)). A key lemma is of independent interest. Let G be the complement of a bipartite graph with partite sets X, Y. If G is 2 k connected, then G contains k Hamilton cycles that are pairwise edge‐disjoint except for edges in G[ Y]. © 2005 Wiley Periodicals, Inc. J Graph Theory 相似文献
19.
In 1985, Thomassen [14] constructed for every positive integer r, finite digraphs D of minimum degree δ( D)= r which do not contain a vertex x lying on three openly disjoint circuits, i.e. circuits which have pairwise exactly x in common. In 2005, Seymour [11] posed the question, whether an r-regular digraph contains a vertex x such that there are r openly disjoint circuits through x. This is true for r≤3, but does not hold for r≥8. But perhaps, in contrast to the minimum degree, a high regularity degree suffices for the existence of a vertex lying on r openly disjoint circuits also for r≥4. After a survey of these problems, we will show that every r-regular digraph with r≥7 has a vertex which lies on 4 openly disjoint circuits. 相似文献
20.
Dedicated to the memory of Paul Erdős
A graph G is k-linked if G has at least 2k vertices, and, for any vertices , , ..., , , , ..., , G contains k pairwise disjoint paths such that joins for i = 1, 2, ..., k. We say that G is k-parity-linked if G is k-linked and, in addition, the paths can be chosen such that the parities of their lengths are prescribed. We prove the existence of a function g( k) such that every g( k)-connected graph is k-parity-linked if the deletion of any set of less than 4 k-3 vertices leaves a nonbipartite graph. As a consequence, we obtain a result of Erdős–Pósa type for odd cycles in graphs
of large connectivity. Also, every -connected graph contains a totally odd -subdivision, that is, a subdivision of in which each edge of corresponds to an odd path, if and only if the deletion of any vertex leaves a nonbipartite graph.
Received May 13, 1999/Revised June 19, 2000 相似文献
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