On Polynomial Solvability of the Hamiltonian Cycle Problem for Graphs of Degree Less Than or Equal To 3

TL;DR

This paper presents a polynomial-time method for determining Hamiltonicity in graphs with degree ≤ 3 by representing graphs as two basic figures with weighted edges, enabling a check based on parameter comparison.

Contribution

It introduces a novel representation of graphs as two basic objects with weighted edges, allowing polynomial-time Hamiltonicity testing for degree ≤ 3 graphs.

Findings

The method correctly identifies Hamiltonian graphs.

The algorithms operate in polynomial time.

The approach extends to graphs of degree up to 3.

Abstract

Any graph can be represented pictorially as a figure. Moreover, it can be represented as two or more figures that can be have different properties to each other. For the purpose of HCP, we represent a graph by two such figures. In each of them, there is an exterior part called the contour, and an interior part. These two figures differ from each other by the constitution of the edges in the interior part. That is, any edges in the interior part for one figure are not in the interior for the other figure. We call these two figures basic objects. We develop rules and algorithms that allow us to represent any graph of degree d <= 3 by two basic objects. Individually, neither of these representations possess the features to easily determine the Hamiltonicity of the graph. However, the combination of these two figures, once certain weights are assigned to their edges, allows us to determine the Hamiltonicity with a polynomial-time check. The rules for the assignment of weights are: 1. The weight of any edge of the interior part is 0, for both objects. 2. In both figures any common edge of the contour has the same weight. The weights of the edges allow us to extend the number of parameters of the objects, that is sufficient to determine the Hamiltonicity of the graph. Then, if the graph is Hamiltonian, then both figures possess the same set of parameters. If the sets of parameters for two figures are different, then the graph is not Hamiltonian. The parameters that determine the Hamiltonicity of the graph are the sums of weights of edges and windows of contours in the figure. The algorithms of their construction do not contain a combinatorial number of elements and have polynomial complexity. We also supply an estimate of the complexity of each algorithm.