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 = pα ∏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 sh...

A perfect dissection of a polygon P into a polygon P ′ is a decomposition of P into internally disjoint pairwise incongruent polygons all similar to P ′. It is known that there is no perfect dissection of an equilateral triangle into smaller equilateral triangles. On the other hand, an equilateral triangle has trivial perfect dissections into any number n of right triangles where n ≥ 3. We give...

A binary extended 1-perfect code C folds over its kernel via the Steiner quadruple systems associated with its codewords. The resulting folding, proposed as a graph invariant for C, distinguishes among the 361 nonlinear codes C of kernel dimension κ obtained via Solov’eva-Phelps doubling construction, where 9 ≥ κ ≥ 5. Each of the 361 resulting graphs has most of its nonloop edges expressible in...

We introduce a family of graphs, called cellular, and consider the problem of enumerating their perfect matchings. We prove that the number of perfect matchings of a cellular graph equals a power of 2 tiroes the number of perfect matchings of a certain subgraph, called the core of the graph. This yields, as a special case, a new proof of the fact that the Aztec diamond graph of order n introduc...

The computation of template matrices is the bottleneck of simple algorithms for perfect phylogeny haplotyping and for perfect phylogeny under mutation and constrained recombination. The fastest algorithms known so far compute them in O(nm) time. In this paper, we describe an algorithm for computing template matrices in O(nm/ log(n)) time. We also present and discuss a conjecture that implies an...

A binary extended 1-perfect code C folds over its kernel via the Steiner quadruple systems associated with its codewords. The resulting folding, proposed as a graph invariant for C, distinguishes among the 361 nonlinear codes C of kernel dimension κ obtained via Solov’eva-Phelps doubling construction, where 9 ≥ κ ≥ 5. Each of the 361 resulting graphs has most of its nonloop edges expressible in...

The question of perfect state transfer existence in quantum spin networks based on weighted graphs has been recently presented by many authors. We give a simple condition for characterizing weighted circulant graphs allowing perfect state transfer in terms of their eigenvalues. This is done by extending the results about quantum periodicity existence in the networks obtained by Saxena, Severini...

Certain subgraphs of a given graph G restrict the minimum number χ(G) of colors that can be assigned to the vertices of G such that the endpoints of all edges receive distinct colors. Some of such subgraphs are related to the celebrated Strong Perfect Graph Theorem, as it implies that every graph G contains a clique of size χ(G), or an odd hole or an odd anti-hole as an induced subgraph. In thi...

We present a simple and e cient dictionary with worst case constant lookup time, equaling the theoretical performance of the classic dynamic perfect hashing scheme of Dietzfelbinger et al. (Dynamic perfect hashing: Upper and lower bounds. SIAM J. Comput., 23(4):738 761, 1994). The space usage is similar to that of binary search trees, i.e., three words per key on average. The practicality of th...

A graph is Berge if no induced subgraph of it is an odd cycle of length at least five or the complement of one. In joint work with Robertson, Seymour, and Thomas we recently proved the Strong Perfect Graph Theorem, which was a conjecture about the chromatic number of Berge graphs. The proof consisted of showing that every Berge graph either belongs to one of a few basic classes, or admits one o...