On the crossing numbers of join products of five graphs of order six with the discrete graph

The main purpose of this article is broaden known results concerning crossing numbers for join of graphs of order six. We give the crossing number of the join product G + Dn, where the graph G consists of one 5-cycle and of one isolated vertex, and Dn consists on n isolated vertices. The proof is done with the help of software that generates all cyclic permutations for a given number k, and creates a new graph COG for calculating the distances between all vertices of the graph. Finally, by adding some edges to the graph G, we are able to obtain the crossing numbers of the join product with the discrete graph Dn and with the path Pn on n vertices for other two graphs.

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On the Crossing Numbers of the Joining of a Specific Graph on Six Vertices with the Discrete Graph

Symmetry ◽

10.3390/sym12010135 ◽

2020 ◽

Vol 12 (1) ◽

pp. 135

Author(s):

Michal Staš

Keyword(s):

Crossing Number ◽

Crossing Numbers ◽

The Individual ◽

Discrete Graph

In the paper, we extend known results concerning crossing numbers of join products of small graphs of order six with discrete graphs. The crossing number of the join product G ∗ + D n for the graph G ∗ on six vertices consists of one vertex which is adjacent with three non-consecutive vertices of the 5-cycle. The proofs were based on the idea of establishing minimum values of crossings between two different subgraphs that cross the edges of the graph G ∗ exactly once. These minimum symmetrical values are described in the individual symmetric tables.

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Determining Crossing Number of Join of the Discrete Graph with Two Symmetric Graphs of Order Five

Symmetry ◽

10.3390/sym11020123 ◽

2019 ◽

Vol 11 (2) ◽

pp. 123 ◽

Cited By ~ 1

Author(s):

Michal Staš

Keyword(s):

Crossing Number ◽

Crossing Numbers ◽

Symmetric Graphs ◽

Isolated Vertex ◽

Minimum Numbers ◽

Discrete Graph ◽

Disconnected Graph

The main aim of the paper is to give the crossing number of the join product G + D n for the disconnected graph G of order five consisting of one isolated vertex and of one vertex incident with some vertex of the three-cycle, and D n consists of n isolated vertices. In the proofs, the idea of the new representation of the minimum numbers of crossings between two different subgraphs that do not cross the edges of the graph G by the graph of configurations G D in the considered drawing D of G + D n will be used. Finally, by adding some edges to the graph G, we are able to obtain the crossing numbers of the join product with the discrete graph D n and with the path P n on n vertices for three other graphs.

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Cyclic permutations and crossing numbers of join products of symmetric graph of order six

Carpathian Journal of Mathematics ◽

10.37193/cjm.2018.02.03 ◽

2018 ◽

Vol 34 (2) ◽

pp. 143-155

Author(s):

STEFAN BEREZNY ◽

◽

MICHAL STAS ◽

Keyword(s):

Crossing Number ◽

Symmetric Graph ◽

Combinatorial Properties ◽

Crossing Numbers ◽

Join Of Graphs ◽

Discrete Graph

In the paper, we extend known results concerning crossing numbers for join of graphs of order six. We give the crossing number of the join product G+Dn, where the graph G consists of two leaves incident with two opposite vertices of one 4-cycle, and Dn consists on n isolated vertices. The proof is done with the help of software that generates all cyclic permutations for a given number k, and creates a new graph COG for a calculating the distances between all (k − 1)! vertices of the graph. Finally, by adding new edges to the graph G, we are able to obtain the crossing number of the join product with the discrete graph Dn for two other graphs. The methods used in the paper are new, and they are based on combinatorial properties of cyclic permutations.

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Determining crossing numbers of the join products of two specific graphs of order six with the discrete graph

Filomat ◽

10.2298/fil2009829s ◽

2020 ◽

Vol 34 (9) ◽

pp. 2829-2846

Author(s):

Michal Stas

Keyword(s):

Connected Graph ◽

Crossing Number ◽

Topological Approach ◽

Crossing Numbers ◽

Minimum Numbers ◽

Discrete Graph

The main aim of the paper is to give the crossing number of the join product G* + Dn for the connected graph G* of order six consisting of P4 + D1 and of one leaf incident with some inner vertex of the path P4 on four vertices, and where Dn consists of n isolated vertices. In the proofs, it will be extend the idea of the minimum numbers of crossings between two different subgraphs from the set of subgraphs which do not cross the edges of the graph G* onto the set of subgraphs by which the edges of G* are crossed exactly once. Due to the mentioned algebraic topological approach, we are able to extend known results concerning crossing numbers for join products of new graphs. The proofs are done with the help of software that generates all cyclic permutations for a given number k, and creates a new graph COG for calculating the distances between all (k-1)! vertices of the graph. Finally, by adding one edge to the graph G*, we are able to obtain the crossing number of the join product of one other graph with the discrete graph Dn.

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The Crossing Numbers of Join Products of Paths and Cycles with Four Graphs of Order Five

Mathematics ◽

10.3390/math9111277 ◽

2021 ◽

Vol 9 (11) ◽

pp. 1277

Author(s):

Michal Staš

Keyword(s):

Connected Graph ◽

Crossing Numbers ◽

Tripartite Graph ◽

Complete Tripartite Graph

The main aim of the paper is to establish the crossing numbers of the join products of the paths and the cycles on n vertices with a connected graph on five vertices isomorphic to the graph K1,1,3\e obtained by removing one edge e incident with some vertex of order two from the complete tripartite graph K1,1,3. The proofs are done with the help of well-known exact values for the crossing numbers of the join products of subgraphs of the considered graph with paths and cycles. Finally, by adding some edges to the graph under consideration, we obtain the crossing numbers of the join products of other graphs with the paths and the cycles on n vertices.

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