Peter Horák

Induced matchings in cubic graphs

In this paper, we show that the edge set of a cubic graph can always be partitioned into 10 subsets, each of which induces a matching in the graph. This result is a special case of a general conjecture made by Erdos and NeSetiil: For each d 2 3, the edge set of a graph of maximum degree d can always be partitioned into [5d2/4] subsets each of which induces a matching. 0 1993 John Wiley & Sons, Inc.

The train marshalling problem

Abstract The problem considered in this paper arose in connection with the rearrangement of railroad cars in China. In terms of sequences the problem reads as follows: Train Marshalling Problem: Given a partition S of {1,…,n} into disjoint sets S1,…,St, find the smallest number k=K(S) so that there exists a permutation p(1),…,p(t) of {1,…,t} with the property: The sequence of numbers 1,2,…,n,1,2,…,n,…,1,2,…,n where the interval 1,2,…,n is repeated k times contains all the elements of Sp(1), then all elements of S p(2) ,… , etc., and finally all elements of Sp(t). The aim of this paper is to show that the decision problem: “Given numbers n,k and a partition S of {1,2,…,n}, is K(S)⩽k?” is NP-complete. In light of this, we give a general upper bound for K(S) in terms of n.

On perfect Lee codes

In this paper we survey recent results on the Golomb-Welch conjecture and its generalizations and variations. We also show that there are no perfect 2-error correcting Lee codes of block length 5 and 6 over Z. This provides additional support for the Golomb Welch conjecture as it settles the two smallest cases open so far.

Tilings in Lee metric

Gravier et al. proved [S. Gravier, M. Mollard, Ch. Payan, On the existence of three-dimensional tiling in the Lee metric, European J. Combin. 19 (1998) 567-572] that there is no tiling of the three-dimensional space R^3 with Lee spheres of radius at least 2. In particular, this verifies the Golomb-Welch conjecture for n=3. Spacapan, [S. Spacapan, Non-existence of face-to-face four-dimensional tiling in the Lee metric, European J. Combin. 28 (2007) 127-133], using a computer-based proof, showed that the statement is true for R^4 as well. In this paper we introduce a new method that will allow us not only to provide a short proof for the four-dimensional case but also to extend the result to R^5. In addition, we provide a new proof for the three-dimensional case, just to show the power of our method, although the original one is more elegant. The main ingredient of our proof is the non-existence of the perfect Lee 2-error correcting code over Z of block size n=3,4,5.

On Alspach's conjecture

Publisher Summary This chapter describes on Alspachs conjecture. The chapter shows that if all cycles are of length 3, 4 or 6, and if n is odd and 3a + 4b + 6c = n(n - 1)/2 , or if n is even and 3a + 4b + 6c = n(n - 2)/2 , then G = aC 3 + bC 4 + cC 6 where G = K n if n is odd and G = K n - F if n is even. Some theorems and their proofs are also described in the chapter.

The Prism Over the Middle-levels Graph is Hamiltonian

Let Bk be the bipartite graph defined by the subsets of {1,…,2k + 1} of size k and k + 1. We prove that the prism over Bk is hamiltonian. We also show that Bk has a closed spanning 2-trail.

Diameter Perfect Lee Codes

Lee codes have been intensively studied for more than 40 years. Interest in these codes has been triggered by the Golomb-Welch conjecture on the existence of the perfect error-correcting Lee codes. In this paper, we deal with the existence and enumeration of diameter perfect Lee codes. As main results, we determine all q for which there exists a linear diameter-4 perfect Lee code of word length n over Zq, and prove that for each n ≥ 3, there are uncountable many diameter-4 perfect Lee codes of word length n over Z. This is in a strict contrast with perfect error-correcting Lee codes of word length n over Z as there is a unique such code for n=3, and its is conjectured that this is always the case when 2n+1 is a prime. We produce diameter perfect Lee codes by an algebraic construction that is based on a group homomorphism. This will allow us to design an efficient algorithm for their decoding. We hope that this construction will turn out to be useful far beyond the scope of this paper.

A combinatoral problem in database security

Abstract Let A be a K-dimensional matrix of size d1 × … × dk. By a contiguous submatrix B of A we understand the matrix B = {ainil…ik}, il … ik ϵ Il × … × lk, where Is is an interval, Is ⊂ l, …, ds, s = l, …, k. For a contiguous submatrix B we denote by SUM(B) the sum of all elements of B. The following question has been raised in connection with the security of statistical databases. What is the largest family B of contiguous submatrices of A so that knowing the value of SUM(B) for all B in B does not enable one to calculate any of the elements of A? In this paper we show that, for all k, the largest set B is uniquely determined and equals the set of all contiguous submatrices with an even number of elements of A.

Minimal Oriented Graphs of Diameter 2

Abstract. Let f(n) be the minimum number of arcs among oriented graphs of order n and diameter 2. Here it is shown for n>8 that (1−o(1))n log n≤f(n)≤n log n−(3/2)n.

Kirkman's school projects

Abstract We discuss the generalization of Kirkmans Schoolgirl Problem to the case where the number of schoolgirls is not a multiple of 3. It is required that all blocks be of size 3 except that there may be one block per round of size 2, or one of size 4.