William Kocay

On Writing Isomorphism Programs

This is a self-contained exposition on how to write isomorphism programs. It is intended for people who want to write isomorphism programs for combinatorial structures, such as graphs, designs, digraphs, posets, etc.

Some NP-complete problems for hypergraph degree sequences

Abstract A k-hypergraph G has vertex-set V(G) and edge-set E(G) consisting of k-subsets of V(G). If uϵV(G), then those edges of G containing u define a (k−1)-hypergraph Gu. We say G subsumes the (k−1)-hypergraphs {Gu|;uϵV(G)}. Given n graphs (i.e., 2-hyperegraphs) g1, g2, … gn, is there a 3-hypergraph G such that the subsumed graphs G i ≽g i , for i=1, 2, …, n? Given only the degree sequences of n graphs g1, g2, …, gn, is there a 3-hypergraph G whose subsumed graphs G1, G2, …, Gn have the same degree sequences? We consider 3-hypergraphs with and without repeated edges. We prove these problems NP-complete. We indicate their relation to some well-known problems. The corresponding problems for 2-hypergraphs have simple polynomial solutions.

A family of nonreconstructible hypergraphs

A k-hypergraph is a hypergraph in which each edge contains k vertices. We describe the construction of an infinite family of finite, nonreconstructible 3-hypergraphs. We also indicate why the same techniques can likely be used to construct nonreconstructible k-hypergraphs for any k ⩾ 3.

An extension of the multi-path algorithm for finding Hamilton cycles

The multi-path algorithm for finding hamilton cycles in a graph G is described in the book by Christofides. It is an intelligent exhaustive search for a hamilton cycle. In this paper we describe how the algorithm can be improved in two ways: (1) by detecting small separating sets M for which G‐M has more than |M| components; and (2) by detecting bipartitions (X,Y), where |X|<|Y|. Let G be an undirected graph with n vertices V(G), and e edges E(G). We consider only simple graphs (that is, no loops or multiple edges), so that every edge is uniquely defined by a pair of vertices. If u and v are vertices, then the pair {u,v} is written as uv. It is often convenient to write u → v if u is adjacent to v, that is uv ∈E(G), and u → / v if u is not adjacent to v (cf. Hopcroft and Tarjan [4]). We want to find a hamilton cycle in G, or to determine that G is non-hamiltonian. We assume throughout that G is a 2-connected graph, since G cannot be hamiltonian if it is not 2-connected. This problem is NP-complete, so we cannot realistically expect to find a polynomial algorithm. Nevertheless, it is a fundamental question in graph theory to determine whether a graph is hamiltonian, and we would like to do so as efficiently as possible, for as wide a range of graphs as possible. In this article, we describe an algorithm based on the “multi-path” method described in Christophides [3]. It is an exhaustive search of all paths in the graph which may extend to hamilton cycles. In general, it is the nonhamiltonian graphs which are difficult, since we must examine the entire search tree, which is equivalent to the problem of finding all hamilton cycles of a hamiltonian graph; whereas with hamiltonian graphs, we can stop as soon as we find one cycle. The algorithm described herein is able to recognize, in certain cases, when the current path cannot possibly extend to a hamilton cycle, and this enables us to prune off portions of the search tree, sometimes very large portions. It may be possible to extend the method to include pruning in other instances as well in order to develop a still more efficient algorithm. 2. The Multi-Path Method.

On strong starters in cyclic groups

Abstract Strong starters have been very useful in the construction of Room squares and cubes, Howell designs, Kirkman triple systems and Kirkman squares and cubes. In this paper we investigate various properties of strong starters in cyclic groups. In particular, we enumerate all non-isomorphic strong starters in cyclic groups of order n ⩽ 23 and all non-equivalent ones of order n⩽ 27. We also obtain results on the automorphism groups of the corresponding 1-factorizations and their embeddibility in Kirkman triple systems.

Reconstructing graphs as subsumed graphs of hypergraphs, and some self-complementary triple systems

AbstractLetV be a set ofn elements. The set of allk-subsets ofV is denoted

More non-reconstructible hypergraphs

\left( {_k^V } \right)

Hypomorphisms, orbits and reconstruction

. Ak-hypergraph G consists of avertex-set V(G) and anedgeset