度小于或等于3的图的哈密顿圈问题的多项式可解性
On Polynomial Solvability of the Hamiltonian Cycle Problem for Graphs of Degree Less Than or Equal To 3
摘要：
任何图形都可以用图形来表示。此外，它可以表示为两个或更多个图形，这些图形可以具有彼此不同的属性。为了HCP的目的，我们用两个这样的图形来表示一个图。在每一个中，都有一个外部部分称为轮廓，还有一个内部部分。这两个图形的不同之处在于内部边缘的构成。也就是说，一个图形的内部部分中的任何边缘不在另一个图形的内部中。我们称这两个图形为基本对象。我们开发的规则和算法，使我们能够表示任何图的度d <= 3的两个基本对象。单独地，这两种表示都不具有容易确定图的哈密顿性的特征。然而，这两个数字的组合，一旦某些权重被分配给他们的边缘，允许我们确定一个多项式时间检查的哈密尔顿性。权重分配的规则是：1。对于这两个对象，内部零件的任何边的权重都为0。2.在两个图中，轮廓的任何公共边缘具有相同的权重。边的权重允许我们扩展对象的参数数量，这足以确定图的Hamilton性。然后，如果图是哈密尔顿的，那么两个图具有相同的参数集。如果两个图的参数集不同，则该图不是哈密尔顿图。决定图的哈密尔顿性的参数是图中轮廓的边和窗口的权重之和。其构造的算法不包含元素的组合数量，并且具有多项式复杂性。我们还提供了每个算法的复杂性的估计。
英文摘要：
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.
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