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The article addresses algebraic methods for coloring arbitrary cubic graphs. The results are partially based on the corollaries of the Tait theorem. In the article, the authors propose using a fourth-order Klein group transform in order to formally describe the coloring of a cubic graph. The transition to graph coloring is done by coloring the edges of basis cycles. Overall, the mathematical framework for describing topological graph drawing is presented and formally described in the article. Based on the edge coloring, the formation of colored disks and the mathematical description of the operation of colored disks rotation with subsequent recoloring of the edges are considered. It is shown that the operation of rotating color disks can be represented as a ring sum (addition modulo 2) of cycles. In order to unambiguously describe the representation of colored disks by means of basis cycles, the authors introduce the concept of embeddability of colored disks. For clarity, the authors provide several examples illustrating the application of colored disks rotation operation to concrete cubic graphs. The relation between the system of induced cycles generated by the rotation of graph vertices and the coloring of 2-factors of the cubic graph is established in the present study. It is shown that the ring sum of all cycles included in the colored 2-factors of the graph is an empty set. The article also addresses the issues of coloring non-planar cubic graphs. The relationship between basis cycles and a rim in a non-planar cubic graph and a ring sum of colored 2-factors is explicitly shown in the article. In addition, the relationship between the colored vertex rotation of a plane cubic graph and the closed Heawood paths is revealed and formally described.

Запорожский национальный университет, г. Запорожье, УкраинаРассматривается алгоритм проверки графа на планарность с одновременным по-строением математических структур для описания топологического рисунка плос-кого графа. Такими математическими структурами являются изометрические циклы и вращение вершин графа. Показано, что система изометрических цик-лов графа индуцирует вращение вершин для описания топологического рисунка плоского графа. В отличие от классических алгоритмов проверки планарности, например алгоритма Хопкрофта Тарьяна, полученный в результате работы ал-горитма топологический рисунок используется для визуализации плоского графа. Вычислительная сложность алгоритма определяется как O(m 2 ), где m количе-ство рёбер графа.Ключевые слова: граф, планарность, визуализация графа, топологический ри-сунок графа, алгоритмы на графах, вращение вершин, изометрические циклы. In this article we present a new graph planarity testing algorithm along with the construction of mathematical framework used for representing topological drawings of plane graphs. This mathematical framework is based on the notions of graph isometric cycles and rotation of graph vertices. It is shown that the system of isometric cycles of a graph induces the rotation of its vertices for representing topological drawing of the plane graph. In contrast to the classical planarity testing algorithms, e. g. the Hopcroft -Tarjan algorithm, the topological drawing obtained as a result of the proposed algorithm execution is used subsequently for the visualization of the planar graph. Computational complexity of the proposed algorithm is estimated by O(m 2 ), where m is the number of edges in the graph.