Dean G. Hoffman

A new class of group divisible designs with block size three

Abstract The elementary necessary conditions for the existence of a group divisible design with block size three, t groups of size g , and one group of size u are shown to be sufficient for all choices of g , t , and u .

On the construction of odd cycle systems

Three obvious necessary conditions for the existence of a k-cycle system of order n are that if n > 1 then n ⩾ k, n is odd, and 2k divides n(n − 1). We show that if these necessary conditions are sufficient for all n satisfying k ⩽ n < 3k then they are sufficient for all n. In particular, there exists a 15-cycle system of order n if and only if n ≡ 1, 15, 21, or 25 (mod 30), and there exists a 21-cycle system of order n if and only if n ≡ 1, 7, 15, or 21 (mod 42), n ≠ 7. 15.

The chromatic index of complete multipartite graphs

We show that a complete multipartite graph is class one if and only if it is not eoverfull, thus determining its chromatic index.

A note on a conjecture concerning symmetric resilient functions

Abstract In 1985, Chor et al. conjectured that the only 1-resilient symmetric functions are the exclusive-or of all n variables and its negation. In this note the existence of symmetric resilient functions is shown to be equivalent to the existence of a solution to a simultaneous subset sum problem. Then, using arithmetic properties of certain binomial coefficients, an infinite class of counterexamples to the conjecture is obtained.

The number of repeated blocks in twofold triple systems

Abstract In this paper, we give a complete answer to the following question: Given an integer υ ≡ 0 or 1 (mod 3) and an integer k, does there exist a twofold triple system of order υ with exactly k repeated triples? In particular, we prove the following theorem: If υ ≡ 0 or 4 (mod 6), υ > 12, then there exists a twofold triple system of order υ having exactly k repeated triples if and only if k ϵ I′υ, where I′υ = {0, 1,…, sυ − 2, sυ} if υ ≡ 0 (mod 4), I′υ = {0, 1,…, sυ − 1} if υ ≡ 2 (mod 4), and s υ = υ(υ − 4) 6 .

Blocking sets in designs with block size 4

We construct λ-fold block designs with block size 4 and λ ∈ {1, 2} of every admissable order which admit a blocking set with three possible exceptions for λ =1 and five possible exceptions for λ=2.

Class one graphs

Abstract Conditions on the subgraph induced by the vertices of maximum degree of a simple graph G are found which are sufficient for G to be class 1. These conditions can be used to generalize a result of Fournier. An efficient algorithm is described for deciding whether or not a particular graph satisfies the conditions.

Decomposition of complete tripartite graphs into gregarious 4-cycles

A 4-cycle in a tripartite graph with vertex partition {V1, V2, V3} is said to be gregarious if it has at least one vertex in each Vi, 1 ≤ i ≤ 3. In this paper, necessary and sufficient conditions are given for the existence of an edge-disjoint decomposition of any complete tripartite graph into gregarious 4-cycles.

5-Cycle Systems with Holes

Recently the generalized Doyen-Wilson problem of embedding a 5-cycle system of orderu in one of orderv was completely solved. However it is often useful to solve the more general problem of the existence of a 5-cycle system of orderv with a hole of sizeu. In this paper we completely solve this problem.

Bidegreed graphs are edge reconstructible

An edge-deleted subgraph of a graph G is a subgraph obtained from G by the deletion of an edge. The Edge Reconstruction Conjecture asserts that every simple finite graph with four or more edges is determined uniquely, up to isomorphism, by its collection of edge-deleted subgraphs. A class of graphs is said to be edge reconstructible if there is no graph in the class with four or more edges that is not edge reconstructible. This paper proves that bidegreed graphs (graphs whose vertices all have one of two possible degrees) are edge reconstructible. The results are then generalized to show that all graphs that do not have three consecutive integers in their degree sequence are also edge reconstructible.