Paul J. Schellenberg

The Oberwolfach problem and factors of uniform odd length cycles

Abstract Let m ⩾ 3 be an odd integer. In this paper it is shown that if n ⩾ m is odd and m divides n, then the edge-set of the complete graph Kn can be partitioned into 2-factors each of which is comprised of m-cycles only. Similarly, if n is an even multiple of m, n ≠ 4m and n > 6, then the edge-set of the complete graph on n vertices with a 1-factor removed can also be partitioned into 2-factors each of which is comprised of m-cycles.

The enumeration of c-nets via quadrangulations

Abstract A counting procedure for simple quadrangulations is established. Using the technique of counting simple quadrangulations together with a one-to-one correspondence between simple quadrangulations and c-nets, the enumeration of c-nets with i+1 vertices and j+1 faces is accomplished.

The existence of C k -factorizations of K 2 n - F

Abstract A necessary condition for the existence of a C k -factorization of K 2n − F is that k divides 2 n . It is known that neither K 6 − F nor K 12 − F admit a C 3 -factorization. In this paper we show that except for these two cases, the necessary condition is also sufficient.

On the existence of frames

This paper deals with two topics, namely, frames and pairwise balanced designs (PBDs). Frames, which were introduced by W.D. Wallis for the construction of (skew) Room squares, are shown to exist for most orders congruent to 1 (mod 4). This result relies heavily on the existence of PBDs since the set F = {v | there is a frame of order v] is shown to be PBD-closed. By employing a generalization of the usual recursive construction for PBDs, it is shown that B{5, 9, 13, 17}@?B{5, 9, 13}@?{69, 77, 97, 137, 237, 277, 317, 377, 569}@?{n | n @? 1 (mod 4), n>0}@?{29, 33, 49, 57, 93, 129, 133}, where B(K) denotes the set of orders of PBDs of index one having block-sizes from the set K. Frames of orders 5, 9, 13 and 17 are exhibited which immediately implies that F@?B{5, 9, 13, 17}. D.R. Stinson and W.D. Wallis have shown that {29, 49}@?F. Thus there is a frame of order @u for every positive integer @u congruent to 1 (mod 4) with the possible exceptions of @u @e {33, 57, 93, 133}.

Hanani triple systems

Hanani triple systems onv≡1 (mod 6) elements are Steiner triple systems having (v−1)/2 pairwise disjoint almost parallel classes (sets of pairwise disjoint triples that spanv−1 elements), and the remaining triples form a partial parallel class. Hanani triple systems are one natural analogue of the Kirkman triple systems onv≡3 (mod 6) elements, which form the solution of the celebrated Kirkman schoolgirl problem. We prove that a Hanani triple system exists for allv≡1 (mod 6) except forv ∈ {7, 13}.

Some Results on the Existence of Squares

In order to investigate the spectrum of skew Room squares, the authors obtain some preliminary results in preparation for a recursive attack. Some of these seem to be of interest in their own right. For example, if N(ν) denotes the maximum number of mutually orthogonal Latin squares of side ν, it is shown, by direct construction, that N(82) ⩾ 8 and N(100) ⩾ 8 and, by using recursive constructions, that N (ν)⩾ 8 for ν ⩾ 9445. It is also shown that for ν = 2 α t + 1, (2, t)= 1, aL α ≠ 1, 2, 6, 7, there exists a skew Room square of side ν.

A Parallel Algorithm for Extending Cryptographic Hash Functions

We describe a parallel algorithm for extending a small domain hash function to a very large domain hash function. Our construction can handle messages of any practical length and preserves the security properties of the basic hash function. The construction can be viewed as a parallel version of the well known Merkle-Damgard construction, which is a sequential construction. Our parallel algorithm provides a significant reduction in the computation time of the message digest, which is a basic operation in digital signatures.

On Decomposing Graphs into Isomorphic Uniform 2-Factors

Publisher Summary For v, an even integer, let H v be the complete graph on v vertices with the edges of a 1 - factor deleted and, for v odd, let H v be the complete graph on v vertices. The Oberwolfach problem is to determine whether, for any given 2-factor G of H v , where v is odd, it is possible to decompose H v into 2 - factors, each of which is isomorphic to G. The corresponding problem when v is even is called the spouse-avoiding Oberwolfach problem. Thus, the chapter investigates special problems of these kinds.

The existence of howell designs of siden+1 and order 2n

AHowell design of side s andorder 2n, or more briefly, anH(s, 2n), is ans×s array in which each cell either is empty or contains an unordered pair of elements from some 2n-set, sayX, such that(a) each row and each column is Latin (that is, every element ofX is in precisely one cell of each row and each column) and(b) every unordered pair of elements fromX is in at most one cell of the array. Atrivial Howell design is anH(s, 0) havingX=Ø and consisting of ans×s array of empty cells. A necessary condition onn ands for the existence of a nontrivialH(s, 2n) is that 0<n≦s≦2n-1.AnH(n+t, 2n) is said to contain a maximum trivial subdesign if somet×t subarray is theH(t, 0). This paper describes a recursive construction for Howell designs containing maximum trivial subdesigns and applies it to settle the existence question forH(n+1, 2n)’s: forn+1 a positive integer, there is anH(n+1, 2n) if and only ifn+1 ∉ {2, 3, 5}.

A construction for equidistant permutation arrays of index one

An equidistant permutation array (EPA) which we denote by A(r, λ; ν) is a ν × r array such that every row is a permutation of the integers 1, 2,…, r and such that every pair of distinct rows has precisely λ columns in common. R(r, λ) is the maximum ν such that there exists an A(r, λ; ν). In this paper we show that R(n2 + n + 2, 1) ⩾ 2n2 + n where n is a prime power.