What is the fewest number of open trails into which the edges of GraphBuster can be partitioned?

Name: Problem 32, Chapter 11: Introductory Combinatorics
Uploaded: 2020-08-18T03:27:54-08:00
Duration: 8 min 32 s
Channel: Chris Trentman
Description: What is the fewest number of open trails into which the edges of GraphBuster can be partitioned?

step 1 We're asked to find the crossing number of the Peterson graph. Now, we have, by the last example

What is the fewest number of open trails into which the...

Key Concepts

Graph Theory

Graph theory is the mathematical study of graphs, which are structures made up of vertices (or nodes) connected by edges. It provides the foundational language and concepts used to analyze problems involving connectivity, traversals, and the overall structure of networks.

Trail

A trail is a sequence of edges in a graph where each edge is used only once. Trails are crucial in understanding how a graph can be traversed without repeating edges, which is a common constraint in various graph traversal problems.

Open Trail

An open trail is a trail where the starting and ending vertices are distinct. This concept contrasts with closed trails or circuits, where the trail begins and ends at the same vertex, and is fundamental in questions of partitioning graphs into trails with specific endpoint conditions.

Eulerian Paths and Circuits

An Eulerian path is a trail that uses every edge of a graph exactly once, and an Eulerian circuit is an Eulerian path that starts and ends at the same vertex. The existence of Eulerian paths or circuits is tightly linked to the parity of vertex degrees, and when a graph is not Eulerian, strategies such as edge partitioning into open trails are applied.

Vertex Degree and Parity

Vertex degree refers to the number of edges incident to a vertex. The parity (whether the degree is odd or even) plays a critical role in determining the traversal properties of the graph, including the possibility of forming Eulerian paths or circuits. When there are more than two vertices with an odd degree, partitioning the graph into open trails typically involves matching these odd degree vertices.

Transcript

00:01 We're asked to find the crossing number of the peterson graph.

00:08 Now, we have, by the last example in this section, that pearson's graph is not planar.

00:19 So it follows that pearson's graph has a crossing number must be greater than or equal to 1.

00:40 Now, suppose you remove an edge from the graph.

01:05 The graph will not become planar because the graph will still contain a subgraph homeomorphic to k33.

01:42 So for example, if the edge that was removed was not between a vertex of the set fdj and a vertex of the set eih, then by the same reason in the textbook we have that the graph still contains a subgroup, subgraph homeomorphic to k -33, and otherwise we can just rename one of the other vertices so that again we can use.

02:18 Use the statement in the example.

02:24 So this means that the crossing number graph is non -planner still.

02:49 And so the new graph has a crossing number greater than or equal to 1.

03:02 And so it follows that peterson graph has a crossing number greater than or equal to 2, since we're adding back an edge.

03:29 And now, let's move the vertical of peterson's graph so we can obtain exactly two crossings.

03:40 So for example, i could move a and b inside as well as e inside.

04:06 So i'll draw the new pearson graph down here.

04:23 We'll keep c and d in the same place.

04:45 But now let's move f up above a.

04:58 Well, we have the f is going to be connected to a.

05:32 And let's put g in the middle of a, b, and e...