Visual algorithm for coloring planar graphs

Kurapov, S.V.; Davidovsky, Maxim; Tolok, A. V.

doi:10.26583/sv.10.3.01

Cited by 2 publications

(6 citation statements)

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“…Next, we form the rim of the graph, c0 = {e18 [1],e19 [3],e20 [3],e21 [1],e22 [2],e29 [2],e36[3]}. A rule for determining the composition of disks after rotation.…”

Section: Rotation Of Colored Disksmentioning

confidence: 99%

“…For the colored cubic graph Н3 (Fig. 11), we select the following basis cycles: Next, form the rim, c0 = {e1 [3],e3 [1],e5 [2],e13 [3],e14 [2],e19 [3],e20 [1],e21 [3]…”

Section: Rotation Of Colored Disksmentioning

confidence: 99%

“…3  = <e23 [1],e12 [2],e9 [3],e8 [1],e2 [2],e1 [3],e4 [1],e7 [2],e15 [3],e17 [1],e22 [2],e21 [3]>; 4  = <e11 [1],e12 [2],e24 [3],e17 [1],e16 [2],e19 [3],e18 [1],e7 [2],e6 [3],e8 [1],e10 [2],e13[3]>.…”

Section: Colored Rotation Of Verticesmentioning

confidence: 99%

“…In [3], a theorem was proved on the existence of a colored disk passing through linked edges in a basis simple cycle of a planar correctly colored cubic graph H. Based on the proved theorem, it was shown that the four-color problem can be represented as its consequence. The introduction of a new operation -the rotation of a colored disk -made it possible to recolor the edges in a plane colored cubic graph and construct a visual coloring algorithm.…”

Section: Introductionmentioning

confidence: 99%

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Algebraic methods for coloring cubic graphs

Kurapov¹,

Davidovsky²,

Tolok³

2020

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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.

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“…Next, we form the rim of the graph, c0 = {e18 [1],e19 [3],e20 [3],e21 [1],e22 [2],e29 [2],e36[3]}. A rule for determining the composition of disks after rotation.…”

Section: Rotation Of Colored Disksmentioning

confidence: 99%

“…For the colored cubic graph Н3 (Fig. 11), we select the following basis cycles: Next, form the rim, c0 = {e1 [3],e3 [1],e5 [2],e13 [3],e14 [2],e19 [3],e20 [1],e21 [3]…”

Section: Rotation Of Colored Disksmentioning

confidence: 99%

Section: Colored Rotation Of Verticesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Algebraic methods for coloring cubic graphs

Kurapov¹,

Davidovsky²,

Tolok³

2020

Self Cite

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show abstract

“…The graph theory is a fundamental part of the discrete mathematics [1], [2], [3]. Many real problems can be modeled with graphs [4], [5] and others can be described with graph structures [6]. Different problems, such as finding the shortest paths [7], schedule theory problems [8], and many others (including NP-hard problems) can be described with graphs and solved by the corresponding algorithms [9].…”

Section: Introductionmentioning

confidence: 99%

Visual Analysis of Actions Performed with Big Graphs

Kralev*,

Kraleva

2019

IJITEE

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The basic concepts of using application development environments are presented in this paper. The way of using the GraphAnalyser application and its basic functions is also presented. All results of the experiments conducted are generated with this application. According to the experimental methodology, they fall into two groups: the first one includes actions related to the vertices of a graph, and the second one includes actions related to the edges and the dynamic allocation of memory to store the structure of a graph. The results show that when the number of vertices in a graph increases linearly, the time to add and remove these vertices also increases linearly. When the number of graph vertices increases linearly, the number of added vertices per millisecond remains relatively constant. However, the number of vertices removed for one millisecond for graphs containing between 10 and 70 million vertices varies. Similarly, when the number of graph edges increases linearly, the number of added edges per millisecond remains relatively constant. The summarized results of the two experiments show that the actions associated with adding, removing, and calculating the edge lengths are performed much more slowly than adding and removing the vertices.

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Visual algorithm for coloring planar graphs

Cited by 2 publications

References 0 publications

Algebraic methods for coloring cubic graphs

Algebraic methods for coloring cubic graphs

Visual Analysis of Actions Performed with Big Graphs

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