Exploring the Crossing Numbers of Three Join Products of 6-Vertex Graphs with Discrete Graphs
The significance of searching for edge crossings in graph theory lies inter alia in enhancing the clarity and readability of graph representations, which is essential for various applications such as network visualization, circuit design, and data representation. This paper focuses on exploring the...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-05-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/10/1694 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | The significance of searching for edge crossings in graph theory lies inter alia in enhancing the clarity and readability of graph representations, which is essential for various applications such as network visualization, circuit design, and data representation. This paper focuses on exploring the crossing number of the join product <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>G</mi><mo>*</mo></msup><mo>+</mo><msub><mi>D</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>G</mi><mo>*</mo></msup></semantics></math></inline-formula> is a graph isomorphic to the path on four vertices <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>4</mn></msub></semantics></math></inline-formula> with an additional two vertices adjacent to two inner vertices of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mn>4</mn></msub></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mi>n</mi></msub></semantics></math></inline-formula> is a discrete graph composed of <i>n</i> isolated vertices. The proof is based on exact crossing-number values for join products involving particular subgraphs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mi>k</mi></msub></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>G</mi><mo>*</mo></msup></semantics></math></inline-formula> with discrete graphs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>D</mi><mi>n</mi></msub></semantics></math></inline-formula> combined with the symmetrical properties of graphs. This approach could also be adapted to determine the unknown crossing numbers of two other 6-vertices graphs obtained by adding one or two additional edges to the graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>G</mi><mo>*</mo></msup></semantics></math></inline-formula>. |
|---|---|
| ISSN: | 2227-7390 |