Question

construct a graph with vertices \\(r, s, t, u, v, w\\) that has an euler circuit and the degree of \\(w\\) is 4.

what is the edge set?

Explanation:

Identify the graph requirements

We need a connected graph with:

• Vertices: \(V = \{R, S, T, U, V, W\}\) (6 vertices).
• An Euler circuit: This requires the graph to be connected and every vertex to have an even degree.
• The degree of vertex \(W\) must be exactly 4: \(\deg(W) = 4\).

Design a valid degree sequence

To satisfy the Euler circuit condition, all vertices must have even degrees:

• \(\deg(W) = 4\)
• Let the other 5 vertices \(R, S, T, U, V\) have degree 2.
• A simple way to achieve this is to construct a cycle containing all vertices, and then add two more edges incident to \(W\) to increase its degree to 4 while keeping all other degrees even.

Construct the edges

Let's start with a single cycle containing all 6 vertices:

• Cycle: \(R - S - T - U - V - W - R\)
• This gives initial edges: \(\{R, S\}, \{S, T\}, \{T, U\}, \{U, V\}, \{V, W\}, \{W, R\}\).
• Currently, every vertex has degree 2.

To increase \(\deg(W)\) to 4 while keeping all other degrees even:

• We can add a second cycle that shares vertex \(W\).
• For example, add the cycle \(S - W - U - S\).
• This adds edges: \(\{S, W\}, \{W, U\}, \{U, S\}\).
• Let's check the new degrees:
• \(\deg(R) = 2\) (edges: \(\{R, S\}, \{W, R\}\))
• \(\deg(S) = 4\) (edges: \(\{R, S\}, \{S, T\}, \{S, W\}, \{U, S\}\))
• \(\deg(T) = 2\) (edges: \(\{S, T\}, \{T, U\}\))
• \(\deg(U) = 4\) (edges: \(\{T, U\}, \{U, V\}, \{W, U\}, \{U, S\}\))
• \(\deg(V) = 2\) (edges: \(\{U, V\}, \{V, W\}\))
• \(\deg(W) = 4\) (edges: \(\{V, W\}, \{W, R\}, \{S, W\}, \{W, U\}\))
• All vertices have even degrees (2 or 4), and the graph is connected. Thus, it contains an Euler circuit, and \(\deg(W) = 4\).

List the final edge set

The constructed edge set \(E\) is:

$$ E = \{\{R, S\}, \{S, T\}, \{T, U\}, \{U, V\}, \{V, W\}, \{W, R\}, \{S, W\}, \{W, U\}, \{U, S\}\} $$

Answer:

A valid edge set for the constructed graph is:

$$ E = \{\{R, S\}, \{S, T\}, \{T, U\}, \{U, V\}, \{V, W\}, \{W, R\}, \{S, W\}, \{W, U\}, \{U, S\}\} $$