Embedding of the join of P3 and C4

  • Thread starter Solarmew
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In summary, the objective is to prove that the join of P3 and C4 has a genus of 1, making it possible to embed on a torus. To achieve this, the picture must be adjusted so that at most two edges cross at any point, and overpasses must be erected at every point where two edges of G cross. The minimum number of overpasses needed for this is the genus of G. However, rearranging the vertices to have only one point of intersection is proving to be a challenge.
  • #1
Solarmew
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Given a join of P3 and C4, "Adjust the picture if necessary so that at most two edges cross in any point (not representing a vertex). Then erect an overpass at every point where two edges of G cross. The genus of G, is the minimum number of overpasses that must be added to the plane so that G can be embedded in the resulting surface."
The objective is to prove that the graph of the join has genus=1, so it can be embedded on a torus. But I can't figure out how to rearrange the vertices in such a way :( I'm assuming there supposed to be only one point of intersection, but the best I can do is three...
Here P3 is 4-6-1, and C4 is 3-7-2-5, but I don't think that's even relevant.
Any help is appreciated.

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nm, i got it...
 

1. What is the "join" of P3 and C4?

The "join" of P3 and C4 refers to the construction of a new graph by connecting each vertex of P3 (a path with 3 vertices) to each vertex of C4 (a cycle with 4 vertices). This results in a graph with 12 vertices and 12 edges.

2. What is the purpose of embedding the join of P3 and C4?

Embedding the join of P3 and C4 allows for the visualization and study of the interactions between the two graphs. It also provides a way to represent more complex structures and patterns in a simpler and more organized manner.

3. How is the join of P3 and C4 represented in an embedding?

In an embedding, the join of P3 and C4 is typically represented as a single graph with the vertices and edges of both P3 and C4 connected in a way that preserves their original structure and relationships.

4. What are some real-world applications of the join of P3 and C4?

The join of P3 and C4 has applications in computer science, specifically in the design and analysis of algorithms. It can also be used in social network analysis to study how different groups interact with each other.

5. Are there other types of joins besides the join of P3 and C4?

Yes, there are other types of joins in mathematics and computer science, such as the join of two graphs, the join of two sets, and the join of two topological spaces. Each type of join has its own properties and applications.

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