In the Perfect Completion problem one wishes to add the fewest possible edges to a graph in order to obtain a perfect graph. How large can the size of the added edge set be compared to the size of the edge set of the original graph? We show that with high probability the smallest perfect graph containing a random graph G(n; n 0:6) has (n 1:8) edges. We also show that with high probability the m...

Suppose g ≥ 2. A natural number N is called a repdigit in base g if it has the shape a g −1 g−1 for some 1 ≤ a < g, i.e., if all of its digits in its base g expansion are equal. The number N is called perfect if σ(N) = 2N , where σ(N) := � d|N d is the usual sum of divisors function. We show that in each base g, there are at most finitely many repdigit perfect numbers, and the set of all such n...

An r-perfect code of a graph G = (V, E) is a set C ⊆ V such that the r-balls centered at vertices of C form a partition of V . It is proved that the direct product of Cm and Cn (r ≥ 1, m,n ≥ 2r+1) contains an r-perfect code if and only if m and n are each a multiple of (r + 1)2 + r2 and that the direct product of Cm, Cn, and C` (r ≥ 1, m,n, ` ≥ 2r +1) contains an r-perfect code if and only if m...

The ground states of the Bernasconi model are binary ±1 sequences of length N with low autocorrelations. We introduce the notion of perfect sequences, binary sequences with one-valued off-peak correlations of minimum amount. If they exist, they are ground states. Using results from the mathematical theory of cyclic difference sets, we specify all values of N for which perfect sequences exist an...

An (s; n; q; t)-perfect hash family is a set of functions 1 ; 2 ; : : :; s from a set V of cardinality n to a set F of cardinality q with the property that every t-subset of V is injectively mapped into F by at least one of the functions i. The paper shows that the maximum value n s;t (q) that n can take for xed s and t has a leading term that is linear in q if and only if t > s. Moreover, for ...

Let k, v, t be integers such that k ≥ v ≥ t ≥ 2. A perfect hash family PHF(N ; k, v, t) can be defined as an N × k array with entries from a set of v symbols such that every N× t subarray contains at least one row having distinct symbols. Perfect hash families have been studied by over 20 years and they find a wide range of applications in computer sciences and in cryptography. In this paper we...

Let σ(n) denote the sum of positive divisors of the natural number n. Such a number is said to be perfect if σ(n) = 2n. It is well known that a number is even and perfect if and only if it has the form 2p−1(2p − 1) where 2p − 1 is prime. It is unknown whether or not odd perfect numbers exist, although many conditions necessary for their existence have been found. For example, Cohen and Hagis ha...

Let σ(n) denote the sum of positive divisors of the natural number n. Such a number is said to be perfect if σ(n) = 2n. It is well known that a number is even and perfect if and only if it has the form 2p−1(2p − 1) where 2p − 1 is prime. No odd perfect numbers are known, nor has any proof of their nonexistence ever been given. In the meantime, much work has been done in establishing conditions ...

We say that a graph G has a perfect H-packing (also called an H-factor) if there exists a set of disjoint copies of H in G which together cover all the vertices of G. Given a graph H , we determine, asymptotically, the Ore-type degree condition which ensures that a graph G has a perfect H-packing. More precisely, let δOre(H,n) be the smallest number k such that every graph G whose order n is di...