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Volume 1, Issue 1, December 2017, Page: 1-3
On One Class of Undirected Graph
Kochkarev Bagram, Department of Mathematics and Mathematical Modelling, Institute of Mathematics and Mechanics Named After N. I. Lobachevsky, Kazan (Volga Region) Federal University, Kazan, Russia
Sibgatullovich, Department of Mathematics and Mathematical Modelling, Institute of Mathematics and Mechanics Named After N. I. Lobachevsky, Kazan (Volga Region) Federal University, Kazan, Russia
Received: Apr. 1, 2017;       Accepted: Apr. 18, 2017;       Published: May 31, 2017
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.
Keywords
An Undirected Graph, A Simple Path, Hamiltonian Cycle, Polynomial Combinatorial Set, Non-polynomial Combinatorial Set
Kochkarev Bagram, Sibgatullovich, On One Class of Undirected Graph, Control Science and Engineering. Vol. 1, No. 1, 2017, pp. 1-3. doi: 10.11648/j.cse.20170101.11
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