We propose new families of graphs which exhibit quantum perfect state transfer. Our constructions are based on the join operator on graphs, its circulant generalizations, and the Cartesian product of graphs. We build upon the results of Bašić and Petković (Applied Mathematics Letters 22(10):1609-1615, 2009) and construct new integral circulants and regular graphs with perfect state transfer. Mo...

In 1968, Golomb and Welch conjectured that there does not exist perfect Lee code in Z with radius r ≥ 2 and dimension n ≥ 3. Besides its own interest in coding theory and discrete geometry, this conjecture is also strongly related to the degree-diameter problems of abelian Cayley graphs. Although there are many papers on this topic, the Golomb-Welch conjecture is far from being solved. In this ...

Given a set S of 2n points in R d , a perfect matching of S is a set of n edges such that each point of S is a vertex of exactly one edge. The weight of a perfect matching is the sum of the Euclidean lengths of all edges. Rao and Smith have recently shown that there is a constant r > 1, that only depends on the dimension d, such that a perfect matching whose weight is less than or equal to r ti...

The concept of diameter perfect codes, which seems to be a natural generalization of perfect codes (codes attaining the sphere–packing bound) is introduced. This was motivated by the “code–anticode” bound of Delsarte in distance regular graphs. This bound in conjunction with the recent complete solutions of diametric problems in the Hamming graphHq (n) and the Johnson graph J (n, k) gives a sha...

A cake is a metaphor for a heterogeneous, divisible good, such as land. A perfect division of cake is efficient (also called Pareto-optimal), envyfree, and equitable. We give an example of a cake in which it is impossible to divide it among three players such that these three properties are satisfied, however many cuts are made. It turns out that two of the three properties can be satisfied by ...

In 1] Levenshtein and Vinck investigated perfect run{length{limited codes which are capable of correcting single peak shifts. As a basic combinatorial tool for the construction of such codes they introduced the concept of a t{shift N{design, which is deened to be an N{elementary subset H = fh 1 ; : : : h N g of a nite additive Abelian group G, with the property that for any = 1; : : : t and any...

A coloring of a q-ary n-dimensional cube (hypercube) is called perfect if, for every n-tuple x, the collection of the colors of the neighbors of x depends only on the color of x. A Boolean-valued function is called correlation-immune of degree n − m if it takes value 1 the same number of times for each m-dimensional face of the hypercube. Let f = χ S be a characteristic function of a subset S o...

This paper studies non-crossing geometric perfect matchings. Two such perfect matchings are compatible if they have the same vertex set and their union is also non-crossing. Our first result states that for any two perfect matchings M and M ′ of the same set of n points, for some k ∈ O(log n), there is a sequence of perfect matchings M = M0,M1, . . . ,Mk = M , such that each Mi is compatible wi...

We study sufficient l-degree (1 ≤ l < k) conditions for the appearance of perfect and nearly perfect matchings in k-uniform hypergraphs. In particular, we obtain a minimum vertex degree condition (l = 1) for 3-uniform hypergraphs, which is approximately tight, by showing that every 3-uniform hypergraph on n vertices with minimum vertex degree at least (5/9+o(1)) ` n 2 ́ contains a perfect matchi...

The sum of such divisors of N is. denoted by o '(N) , and the number of such divisors by T^\N) . By conventions 1 is an exponential divisor of itself, so that a(l) = 1. The functions j(XN) and o^K^) were introduced in [1] and have been studied in [1] and [2], An integer N is said to be e-perfect whenever o^\N) = 2N9 and g-multiperfect when o^\N) = kN for an integer k > 2. In [1] and [2], severa...