Let x = (x0, . . . , xn−1) be an n-chain, i.e., an n-tuple of non-negative integers < n. Consider the operator s : x 7→ x′ = (x0, . . . , x ′ n−1), where x ′ j represents the number of j’s appearing among the components of x. An n-chain x is said to be perfect if s(x) = x. For example, (2,1,2,0,0) is a perfect 5-chain. Analogously to the theory of perfect, amicable, and sociable numbers, one ca...

Although nontrivial perfect binary codes exist only for length n = 2m−1 with m ≥ 3 and for length n = 23, many problems concerning these codes remain unsolved. Herein, we present solutions to some of these problems. In particular, we show that the smallest nonempty intersection of two perfect codes of length 2m − 1 consists of two codewords, for all m ≥ 3. We also provide a complete solution to...

This article is a continuation of our research on one-to-one correspondence between n -dimensional spectral resolutions and observables lexicographic types quantum structures which started in Dvurečenskij Lachman ( https://doi.org/10.1016/j.fss.2021.05.005 ). There we presented the main properties observables, studied depth characteristic points are crucial for study. Here present body research...

We discuss a relative of the perfect numbers for which it is possible to prove that there are infinitely many examples. Call a natural number n prime-perfect if n and σ(n) share the same set of distinct prime divisors. For example, all even perfect numbers are prime-perfect. We show that the count Nσ(x) of prime-perfect numbers in [1, x] satisfies estimates of the form exp((log x) log log log )...

The Hungarian method is an e cient algorithm for nding a minimal cost perfect matching in a weighted bipartite graph. This paper describes an e cient algorithm for nding all minimal cost perfect matchings. The computational time required to generate each additional perfect matching is O(n(n + m)); and it requires O(n+m) memory storage. This problem can be solved by algorithms for nding the Kth-...

The complement of a graph G is denoted by G. χ(G) denotes the chromatic number and ω(G) the clique number of G. The cycles of odd length at least five are called odd holes and the complements of odd holes are called odd anti-holes. A graph G is called perfect if, for each induced subgraph G of G, χ(G) = ω(G). Classical examples of perfect graphs consist of bipartite graphs, chordal graphs and c...

The Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods. The rst of these three approaches yielded the rst (and to date only) proof of the SPGC; the other two remain promising to consider...

The pre-coloring extension problem consists, given a graph G and a subset of nodes to which some colors are already assigned, in finding a coloring of G with the minimum number of colors which respects the pre-coloring assignment. This can be reduced to the usual coloring problem on a certain contracted graph. We prove that pre-coloring extension is polynomial for complements of Meyniel graphs....