A Θ-graph consists of three internally vertex-disjoint paths joining two vertices (because such a graph looks like Θ). 
To obtain the maximum size of a graph with order n and no Θ-subgraphs, consider such a graph G such that it has a blocks which are isomorphic to K2 and b blocks which are cycles. Express n and the size m of G in terms of a, b and the lengths of such cycles. Derive an upper bound on m and show that this bound can be achieved.
Answer. If n is odd, then the maximum number of edges in an n-vertex graph with no Θ-subgraphs is 23 (n − 1). What is the answer for even n?

To derive the maximum size \( m \) of a graph \( G \) with order \( n \) and no \( \Theta \)-subgraphs, let's start with the definitions and assumptions provided.

### Definitions and Initial Setup
- A \( \Theta \)-graph consists of three internally vertex-disjoint paths joining two vertices.
- Graph \( G \) has \( a \) blocks which are isomorphic to \( K_2 \) and \( b \) blocks which are cycles.
- Let the lengths of these \( b \) cycles be \( c_1, c_2, \ldots, c_b \).

### Expression of \( n \) and \( m \) in Terms of \( a \), \( b \), and Cycle Lengths
Each block \( K_2 \) contributes 2 vertices and 1 edge:
- There are \( 2a \) vertices contributed by the \( a \) \( K_2 \) blocks.
- There are \( a \) edges from these \( K_2 \) blocks.

Each cycle \( C_i \) (of length \( c_i \)) contributes \( c_i \) vertices and \( c_i \) edges:
- There are \( \sum_{i=1}^{b} c_i \) vertices contributed by the \( b \) cycles.
- There are \( \sum_{i=1}^{b} c_i \) edges from these cycles.

The total number of vertices \( n \) is:
\[ n = 2a + \sum_{i=1}^{b} c_i \]

The total number of edges \( m \) is:
\[ m = a + \sum_{i=1}^{b} c_i \]

### Upper Bound on \( m \)
To obtain an upper bound on \( m \), let's consider the constraints imposed by the absence of \( \Theta \)-subgraphs. A graph without \( \Theta \)-subgraphs cannot have any two vertices connected by more than two internally vertex-disjoint paths. This limitation affects how we construct the blocks.

### Constructing the Graph for Maximum \( m \)
One way to construct such a graph while maximizing the number of edges is to use cycles and \( K_2 \) blocks in a specific arrangement.

#### Case for Odd \( n \)
For odd \( n \), it has been derived that:
\[ m_{\max} = \left\lfloor \frac{3}{2} (n - 1) \right\rfloor \]

#### Case for Even \( n \)
For even \( n \), we need to consider a slightly different construction to avoid \( \Theta \)-subgraphs while maximizing edges.

If \( n \) is even, let's denote \( n = 2k \) where \( k \) is an integer. The maximum number of edges can be obtained by forming a cycle of length \( 2k \).

### Constructing the Cycle
Consider a single cycle of length \( 2k \) (i.e., all vertices forming one big cycle). This avoids the \( \Theta \)-subgraphs and maximizes the number of edges:
\[ m = n \]

Therefore, for even \( n \):
\[ m_{\max} = n \]

### Conclusion
For odd \( n \), the maximum number of edges in an \( n \)-vertex graph with no \( \Theta \)-subgraphs is:
\[ m_{\max} = \left\lfloor \frac{3}{2} (n - 1) \right\rfloor \]

For even \( n \), the maximum number of edges in an \( n \)-vertex graph with no \( \Theta \)-subgraphs is:
\[ m_{\max} = n \]

This bound can be achieved by forming a single cycle of length \( n \) for even \( n \).

Do you want details or have any questions?

Here are 5 numbered questions you could ask:
1. How do you construct a \( \Theta \)-free graph for odd \( n \)?
2. Can you explain why cycles avoid \( \Theta \)-subgraphs?
3. How would the construction change if self-loops were allowed?
4. Can this method be extended to hypergraphs?
5. What are the implications of this result in graph theory?

**Tip:** When dealing with combinatorial structures, always check for forbidden configurations to simplify problem-solving strategies.

Explore how to maximize the number of edges in a graph with n vertices and no Θ-subgraphs. Understand the structure of cycle graphs and blocks isomorphic to K2 to achieve the upper bound on edge count. Suitable for advanced undergraduate students and researchers in graph theory.

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