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

Staš, Michal

doi:10.7494/opmath.2020.40.3.383

Cited by 4 publications

(4 citation statements)

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“…Using the idea of a separating cycle, if we consider some subdrawing of the graph G * presented in Figure 3a or Figure 3b, then the value of cr(H 2 + D n ) proofed in [33] Removing all edges of the separating cycle v 3 v 4 v 6 v 3 in Figure 3c produces a good drawing that includes H 3 + D n as a subgraph. Consequently, the result of Theorem 3 also implies at least 6…”

Section: Theorem 1 ([31] Theorem 34) Cr(h 1 +mentioning

confidence: 99%

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Crossing Numbers of Join Product with Discrete Graphs: A Study on 6-Vertex Graphs

Fortes

Staš

2023

Mathematics

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Reducing the number of crossings on graph edges can be useful in various applications, including network visualization, circuit design, graph theory, cartography or social choice theory. This paper aims to determine the crossing number of the join product G*+Dn, where G* is a connected graph isomorphic to K2,2,2∖{e1,e2} obtained by removing two edges e1,e2 with a common vertex and a second vertex from the different partitions of the complete tripartite graph K2,2,2, and Dn is a discrete graph composed of n isolated vertices. The proofs utilize known exact crossing number values for join products of specific subgraphs Hk of G* with discrete graphs in combination with the separating cycles. Similar approaches can potentially estimate unknown crossing numbers of other six-vertex graphs with a larger number of edges in join products with discrete graphs, paths or cycles.

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Section: Theorem 1 ([31] Theorem 34) Cr(h 1 +mentioning

confidence: 99%

“…The exact values for crossing numbers of G + D n for all graphs G of order at most four are given by Klešč and Schrötter [10], and, for some connected graphs G of order five and six, they are also listed in . Additionally, it is worth noting that cr(G + D n ) are only available for certain disconnected graphs G [37][38][39][40].…”

Section: Introductionmentioning

confidence: 99%

Crossing Numbers of Join Product with Discrete Graphs: A Study on 6-Vertex Graphs

Fortes

Staš

2023

Mathematics

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“…The exact values for the crossing numbers of G + D n for all graphs G of order at most four are given by Klešč and Schr ötter [21]. Also, the crossing numbers of the graphs G + D n are known for a lot of graphs G of order five and six [1,5,7,10,11,12,13,15,17,18,19,20,22,23,26,27,29,30,33,34,35,36]. In all these cases, the graph G is connected and contains usually at least one cycle.…”

Section: Introductionmentioning

confidence: 99%

“…In all these cases, the graph G is connected and contains usually at least one cycle. The crossing numbers of the join product G + D n are known only for some disconnected graphs [4,24,25,31,32].…”

Section: Introductionmentioning

confidence: 99%

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

BEREŽNÝ¹,

STAŠ²

2022

CJM

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The crossing number $\mathrm{cr}(G)$ of a graph $G$ is the minimum number of edge crossings over all drawings of $G$ in the plane. In the paper, the crossing number of the join product $G^\ast + D_n$ for the connected graph $G^\ast$ on six vertices consisting of one path on four vertices $P_4$ and two leaves adjacent with the same outer vertex of the path $P_4$ is given, where $D_n$ consists of $n$ isolated vertices. Finally, by adding some edges to the graph $G^\ast$, we obtain the crossing numbers of the join products of other four graphs with $D_n$.

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Parity Properties of Configurations

Staš

2022

Mathematics

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In the paper, the crossing number of the join product G*+Dn for the disconnected graph G* consisting of two components isomorphic to K2 and K3 is given, where Dn consists of n isolated vertices. Presented proofs are completed with the help of the graph of configurations that is a graphical representation of minimum numbers of crossings between two different subgraphs whose edges do not cross the edges of G*. For the first time, multiple symmetry between configurations are presented as parity properties. We also determine crossing numbers of join products of G* with paths Pn and cycles Cn on n vertices by adding new edges joining vertices of Dn.

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

Cited by 4 publications

References 10 publications

Crossing Numbers of Join Product with Discrete Graphs: A Study on 6-Vertex Graphs

Crossing Numbers of Join Product with Discrete Graphs: A Study on 6-Vertex Graphs

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

Parity Properties of Configurations

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