An (n, a, b)-perfect double cube is a b × b × b sized n-ary periodic array containing all possible a × a × a sized n-ary array exactly once as subarray. A growing cube is an array whose cj × cj × cj sized prefix is an (nj, a, cj)-perfect double cube for j = 1, 2, . . ., where cj = n v/3 j , v = a 3 and n1 < n2 < · · ·. We construct the smallest possible perfect double cube (a 256×256×256 sized ...

Let n ≥ 5 be an odd integer and K n the complete graph on n vertices. Let i be an integer with 2 ≤ i ≤ (n − 1)/2. A hamiltonian decomposition H of K n is called i-perfect if the set of the chords at distance i of the hamiltonian cycles in H is the edge set of K n. We show that there exists a 3-perfect hamiltonian decomposition of K n for all odd n ≥ 7.

We show that the number of positive integers $$n\le N$$ such $${\mathbb {Z}}/(n^2+n+1){\mathbb {Z}}$$ contains a perfect difference set is asymptotically $$\frac{N}{\log {N}}$$ .

We study perfect state transfer of quantum walks on signed graphs. Our aim is to show that negative edges are useful for perfect state transfer. First, we show that the signed join of a negative 2-clique with any positive (n, 3)-regular graph has perfect state transfer even if the unsigned join does not. Curiously, the perfect state transfer time improves as n increases. Next, we prove that a s...

Suppose G connected graph on p vertices that contains perfect Then G is said to have property n) if p 2: 2(m + n + 1) and if for each pair of disjoint independent M, N E( G) of m, n there exists a perfect matching P in G such that M S;;;; P and 0. We discuss the circumstances under which E(m, n) =? E(x, y), and prove that (surprisingly) in general E(m, n) does not imply E(m, n-1).

The (k,n)-perfect shuffle, a generalisation of the 2-way perfect shuffle, cuts a deck of kn cards into k equal size decks and interleaves them perfectly with the first card of the last deck at the top, the first card of the second-to-last deck as the second card, and so on. It is formally defined to be the permutation ρk,n : i→ ki (mod kn + 1), i ∈ {1,2, . . . ,kn}. We uncover the cycle structu...

Let σ(n) denote the sum of the positive divisors of n. We say that n is perfect if σ(n) = 2n. Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form N = p ∏k j=1 q 2βj j , where p, q1, · · · , qk are distinct primes and p ≡ α ≡ 1 (mod 4). Define the total number of prime factors of N as Ω(N) := α+ 2 ∑k j=1 βj . Sayers show...

perfect matchings. (A perfect matching or 1-factor is a set of disjoint edges covering all vertices.) This generalizes a result of Voorhoeve [11] for the case k = 3, stating that any 3-regular bipartite graph with 2n vertices has at least ( 4 3) n perfect matchings. The base in (1) is best possible for any k: let αk be the largest real number such that any k-regular bipartite graph with 2n vert...

A perfect sequence has the optimal periodic autocorrelation function where all out-of-phase values are zero, and an orthogonal set has orthogonality that the periodic correlation function for any pair of distinct sequences in the set takes zero at the zero shift. The real-valued perfect sequences of period N = 2 are derived from a real-valued Huffman sequence of length 2 + 1 with ν ≥ n whose ou...

We present a N-dimensional generalization of the two-dimensional block-circulant perfect array construction by [Blake, 2013]. As in [Blake, 2013], the families of N-dimensional arrays possess pairwise good zero correlation zone (ZCZ) cross-correlation. Both constructions use a perfect autocorre-lation sequence with the array orthogonality property (AOP). This paper presents a generalization of ...