Katherine Heinrich

Existence of orthogonal latin squares with aligned subsquares

Abstract It is shown that for both v and n even, v > n > 0, there exists a pair of orthogonal latin squares of order v with an aligned subsquare of order n if and only if v ⩾3 n , v ≠6, n ≠2, 6. This is the final case in showing that the above result is true for all v ≠6 and for all n ≠2, 6. When n =6, the analogous result is obtained for incomplete arrays; the case n =2 having been completed in earlier work.

A simple existence criterion for ( g<f )-factors

We simplify the criterion of Lovasz for the existence of a (g, ƒ)-factor when g < ƒ, or when the graph is bipartite. Moreover, we give a simple direct proof, implying an O((g(V)) · |E|) algorithm, for these cases. We then illustrate the convenience of the new criterion by deriving some old and some new facts about (g,ƒ)-factors and [a,b]-factors.

INCOMPLETE SELF-ORTHOGONAL LATIN SQUARES

We show that for all n > 3k + 1, n # 6, there exists an incomplete self-orthogonal latin square of order n with an empty order k subarray, called an ISOLS(n; k), except perhaps when (n; k) s {(6m + i\2m):i = 2,6}.

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.

Conflict-free access to parallel memories

Abstract Conflict-free access to subsets of array elements is essential for the effective utilization of parallel memories. Skewing schemes for array storage that provide conflict-free access to all entries in r × s and s × r subarrays are developed, using a strong connection between conflict-free skewing schemes and latin squares.

Super-simple designs with v⩽32

Abstract A t-(v,k,λ) design D=(X, B ) is a collection B ={B 1 ,B 2 ,…,B b } of k-subsets (called blocks) of a v-set X (with elements called points) such that every t-subset of X is contained in precisely λ blocks. Let p D = max 1⩽i |B i ∩B j | . Let p ∗ =p ∗ (t,v,k,λ)= min D p D , where D is a t-(v,k,λ) design. A super-simple t-(v,k,λ) design D is one with p D =p ∗ . In this paper we survey known results and present some new super-simple 2-(v,4,λ) designs for v⩽32.

Exact coverings of 2-paths by 4-cycles

Abstract In this paper we give necessary and sufficient conditions for the existence of an exact covering of all of the paths of length two of the complete graph by cycles of length four.

P 4 -decompositions of regular graphs

The linear arboricity la(G) of a graph G is the minimum number of linear forests that partition the edges of G. Akiyama, Exoo, and Harary conjectured that [Δ/2] ≤ la(G) ≤ [Δ+1/2] for any simple graph G with maximum degree Δ. The conjecture has been proved to be true for graphs having Δ = 1, 2, 3, 4, 5, 6, 8, 10. Combining these results, we prove in the article that the conjecture is true for planar graphs having Δ(G) ≠ 7. Several related results assuming some conditions on the girth are obtained as well.

Existence of resolvable path designs

For k ⩾ 2 the complete multigraph λKn has a factorization into isomorphic spanning subgraphs, each component of which is a path of length k - 1 iff n ≡ 0 (mod k) and λk(n - 1) ≡ 0 (mod 2(k - 1)).

Dudeney's round table problem

Abstract A set of Hamilton cycles in the complete graph on n vertices is called a Dudeney set, and denoted D ( n ), if every path of length two lies on exactly one of the cycles. In this paper it is shown that: 1. (a) There is a Dudeney set D ( p + 2) if p is prime and 2 is a generator of the multiplicative subgroup of GF( p ). (b) If there is a Dudeney set D ( n + 1), then there is a Dudeney set D (2 n ). (c) For n ⩽ 50, the only n for which the existence of a Dudeney set D ( n ) remains in doubt are n ϵ {27, 29, 35, 37, 41, 47}.