On One Class of Undirected Graph

Abstract

Introduced the concept of polynomial combinatorial sets in enumerative combinatorics and formulates the problem of finding some element with an easily recognized symptom among elements of a combinatorial set. We build an efficient algorithm to solve this problem. We prove that this algorithm does not fit into the formal definition of an algorithm (e.g. “Turing machine”). It is proved that all NP-complete problems are not polynomial. We consider a countable class of undirected Hamiltonian graphs with an odd number of vertices without loops and multiple edges. We prove one typical feature of such graphs: almost every simple path containing all the vertices of the graph is not Hamiltonian cycle. In other words, in the langage of probability theory, the probability that a randomly selected a simple path in this graph containing all vertices, is Hamiltonian cycle tends to zero with growth of number vertices.