Constructing Union-Free pairs of K-Element subsets

It is proved that one can choose [1/2(n/k)] disjoint pairs of k-element subsets of an n-element set in such a way that the unions of the pairs are all different, supposing that n > n(k)

A coding problem for pairs of subsets

Let $X$ be an $n$--element finite set, $0<k\leq n/2$ an integer. Suppose that $\{A_1,A_2\}$ and $\{B_1,B_2\}$ are pairs of disjoint $k$-element subsets of $X$ (that is, $|A_1|=|A_2|=|B_1|=|B_2|=k$, $A_1\cap A_2=\emptyset$, $B_1\cap B_2=\emptyset$). Define the distance of these pairs by $d(\{A_1,A_2\} ,\{B_1,B_2\})=\min \{|A_1-B_1|+|A_2-B_2|, |A_1-B_2|+|A_2-B_1|\}$. This is the minimum number of elements of $A_1\cup A_2$ one has to move to obtain the other pair $\{B_1,B_2\}$. Let $C(n,k,d)$ be the maximum size of a family of pairs of disjoint subsets, such that the distance of any two pairs is at least $d$. Here we establish a conjecture of Brightwell and Katona concerning an asymptotic formula for $C(n,k,d)$ for $k,d$ are fixed and $n\to \infty$. Also, we find the exact value of $C(n,k,d)$ in an infinite number of cases, by using special difference sets of integers. Finally, the questions discussed above are put into a more general context and a number of coding theory type problems are proposed.Comment: 11 pages (minor changes, and new citations added

Tight Euler tours in uniform hypergraphs - computational aspects

By a tight tour in a $k$-uniform hypergraph $H$ we mean any sequence of its vertices $(w_0,w_1,\ldots,w_{s-1})$ such that for all $i=0,\ldots,s-1$ the set $e_i=\{w_i,w_{i+1}\ldots,w_{i+k-1}\}$ is an edge of $H$ (where operations on indices are computed modulo $s$) and the sets $e_i$ for $i=0,\ldots,s-1$ are pairwise different. A tight tour in $H$ is a tight Euler tour if it contains all edges of $H$. We prove that the problem of deciding if a given $3$-uniform hypergraph has a tight Euler tour is NP-complete, and that it cannot be solved in time $2^{o(m)}$ (where $m$ is the number of edges in the input hypergraph), unless the ETH fails. We also present an exact exponential algorithm for the problem, whose time complexity matches this lower bound, and the space complexity is polynomial. In fact, this algorithm solves a more general problem of computing the number of tight Euler tours in a given uniform hypergraph

Matchings and Hamilton Cycles with Constraints on Sets of Edges

The aim of this paper is to extend and generalise some work of Katona on the existence of perfect matchings or Hamilton cycles in graphs subject to certain constraints. The most general form of these constraints is that we are given a family of sets of edges of our graph and are not allowed to use all the edges of any member of this family. We consider two natural ways of expressing constraints of this kind using graphs and using set systems. For the first version we ask for conditions on regular bipartite graphs $G$ and $H$ for there to exist a perfect matching in $G$, no two edges of which form a $4$-cycle with two edges of $H$. In the second, we ask for conditions under which a Hamilton cycle in the complete graph (or equivalently a cyclic permutation) exists, with the property that it has no collection of intervals of prescribed lengths whose union is an element of a given family of sets. For instance we prove that the smallest family of $4$-sets with the property that every cyclic permutation of an $n$-set contains two adjacent pairs of points has size between $(1/9+o(1))n^2$ and $(1/2-o(1))n^2$. We also give bounds on the general version of this problem and on other natural special cases. We finish by raising numerous open problems and directions for further study.Comment: 21 page

Constructions via Hamiltonian theorems

Demetrovics et al [Design type problems motivated by database theory, J. Statist. Plann. Inference 72 (1998) 149-164] constructed a decomposition of the family of all k-element subsets of an n-element set into disjoint pairs (A,B)(A∩B=θ,|A|=|B|=k) where two such pairs are relatively far from each other in some sense. The paper invented a proof method using a Hamiltonian-type theorem. The present paper gives a generalization of this tool, hopefully extending the power of the method. Problems where the method could be also used are shown. Moreover, open problems are listed which are related to the Hamiltonian theory. In these problems a cyclic permutation is to be found when certain restrictions are given by a family of k-element subsets. © 2005 Elsevier B.V. All rights reserved