Yehuda Roditty

Combinatorial reconstruction problems

Abstract A general technique for tackling various reconstruction problems is presented and applied to some old and some new instances of such problems.

On a Reconstruction Problem for Sequences

It is shown that any word of lengthnis uniquely determined by all itsformula]subwords of lengthk, providedk??167n?+5. This improves the boundk??n/2? given in B. Manvelet al.(Discrete Math.94(1991), 209?219).

On zero-sum partitions and anti-magic trees

We study zero-sum partitions of subsets in abelian groups, and apply the results to the study of anti-magic trees. Extension to the nonabelian case is also given.

Monotone paths in edge-ordered sparse graphs

Abstract An edge-ordered graph is an ordered pair ( G , f ), where G = G ( V , E ) is a graph and f is a bijective function, f : E ( G )→{1,2,…,| E ( G )|}. f is called an edge ordering of G. A monotone path of length k in ( G , f ) is a simple path P k+1 : v 1 ,v 2 ,…,v k+1 in G such that either, f (( v i , v i +1 )) f (( v i +1 , v i +2 )) or f (( v i , v i +1 ))> f (( v i +1 , v i +2 )) for i =1,2,…, k −1. Given an undirected graph G, denote by α ( G ) the minimum over all edge orderings of the maximum length of a monotone path. In this paper we give bounds on α ( G ) for various families of sparse graphs, including trees, planar graphs and graphs with bounded arboricity.

Regular Oberwolfach Problems and Group Sequencings

We deal with Oberwolfach factorizations of the complete graphs Kn and K*n, which admit a regular group of automorphisms. We show that the existence of such a factorization is equivalent to the existence of a certain difference sequence defined on the elements of the automorphism group, or to a certain sequencing of the elements of that group. In the particular case of a hamiltonian factorization of the directed graph K*n which admits a regular group of automorphisms G (|G|=n?1), we have that such a factorization exists if and only if G is sequenceable. We shall demonstrate how the mentioned above (difference) sequences may be used in the construction of such factorizations. We prove also that a hamiltonian factorization of the undirected graph Kn (n odd) which admits a regular group of automorphisms G (|G|=(n?1)/2) exists if and only if n?3 (mod4), without further restrictions on the structure of G.

On the largest tree of given maximum degree in a connected graph

We prove that every connected graph G contains a tree T of maximum degree at most k that either spans G or has order at least kδ(G) + 1, where δ(G) is the minimum degree of G. This generalizes and unifies earlier results of Bermond [1] and Win [7]. We also show that the square of a connected graph contains a spanning tree of maximum degree at most three.

Packing and covering of the complete graph with a graph G of four vertices or less

Abstract The maximal number of pairwise edge disjoint graphs G of four vertices or less, in the complete graph K n , and the minimal number of graphs G of four vertices or less, whose union is K n , are determined.

Reconstructing graphs from their k -edge deleted subgraphs

Let G be a graph with m edges and n vertices. We show that if 2m−k>n! or if 2m>(2n) + k then G is determined by its collection of k-edge deleted subgraphs.

More on vertex-switching reconstruction

Abstract A graph is called s -vertex switching reconstructible ( s -VSR) if it is uniquely defined, up to isomorphism by the multiset of unlabeled graphs obtained by switching of all its s -vertex subsets. Stanley proved that a graph with n vertices is s -VSR if the Krawtchouk polynomial P n s has no even roots. Solving balance equations, introduced in Krasnikov and Roditty ( Arch. Math. ( Basel ) 48 (1987). 458-464) for the switching reconstruction problem, we show that a graph is s -VSR if the corresponding Krawtchouk polynomial has one or two even roots laying far from n /2. As a consequence we prove that graphs with sufficiently large number n of vertices are s -VSR for some values of s about n /2. In particular, all graphs are s -VSR for n − 2 s = 0, 1, 3. and if n ≠ 0 (mod 4), for n − 2 s = 2, 6.

Switching reconstruction and Diophantine equations

Abstract Based on a result of R. P. Stanley (J. Combin. Theory Ser. B 38, 1985, 132–138) we show that for each s ≥ 4 there exists an integer Ns such that any graph with n > Ns vertices is reconstructible from the multiset of graphs obtained by switching of vertex subsets with s vertices, provided n ≠ 0 (mod 4) if s is odd. We also establish an analog of P. J. Kellys lemma (Pacific J. Math., 1957, 961–968) for the above s-switching reconstruction problem.