Let K−
n be a graph obtained from Kn by deleting one edge.
How many subgraphs of K−
n are isomorphic to K
−
5
?


The problem asks how many subgraphs of \( K^-_n \) (a complete graph on \( n \) vertices with one edge deleted) are isomorphic to \( K^-_5 \) (a complete graph on 5 vertices with one edge deleted).

### Step 1: Understand the structures
- \( K_n \) is a complete graph with \( \binom{n}{2} \) edges, and \( K^-_n \) is a graph obtained by removing exactly one edge from \( K_n \), leaving \( \binom{n}{2} - 1 \) edges.
- \( K_5 \) is a complete graph on 5 vertices with \( \binom{5}{2} = 10 \) edges, and \( K^-_5 \) has 9 edges (since one edge is removed).

### Step 2: Subgraph isomorphism to \( K^-_5 \)
We need to count how many subgraphs of \( K^-_n \) are isomorphic to \( K^-_5 \), which involves choosing 5 vertices and ensuring that the subgraph induced on these 5 vertices is \( K^-_5 \).

### Step 3: Choose 5 vertices
First, we can select 5 vertices from the \( n \) vertices of \( K^-_n \) in \( \binom{n}{5} \) ways.

### Step 4: Ensure the subgraph is \( K^-_5 \)
For any set of 5 vertices in \( K^-_n \), the subgraph induced on these 5 vertices will either be \( K_5 \) or \( K^-_5 \). To have a subgraph isomorphic to \( K^-_5 \), the deleted edge in \( K^-_n \) must lie between these 5 vertices.

The number of ways to choose 5 vertices where one specific edge is missing is the number of ways to pick 5 vertices that include the missing edge. The missing edge can be part of \( \binom{5}{2} = 10 \) possible edges in a \( K_5 \), and in \( K^-_5 \), exactly one of these 10 edges is removed.

Thus, if the deleted edge in \( K^-_n \) lies within any of the chosen 5 vertices, the subgraph induced on these vertices will be isomorphic to \( K^-_5 \).

### Step 5: Conclusion
Out of the \( \binom{n}{5} \) ways to choose 5 vertices, the number of valid choices where the deleted edge in \( K^-_n \) lies within those 5 vertices is exactly \( \binom{n}{5} \).

Thus, the number of subgraphs of \( K^-_n \) isomorphic to \( K^-_5 \) is:

\[
\boxed{\binom{n}{5}}
\]

### Additional Questions:
1. How many edges are in \( K_n \) if \( n = 6 \)?
2. What is the difference between \( K^-_n \) and \( K_n \) in terms of edge count?
3. How would the answer change if more than one edge is deleted from \( K_n \)?
4. What are the properties of a subgraph isomorphic to \( K_4 \)?
5. How many edges must be deleted from \( K_5 \) to get a tree?

**Tip:** Understanding subgraph isomorphisms can help in counting structures in larger graphs, which is a key concept in combinatorics and graph theory.

Let K−n be a graph obtained from Kn by deleting one edge. How many subgraphs of K−n are isomorphic to K−5?

This problem explores the number of subgraphs of K−n (a complete graph with one edge removed) that are isomorphic to K−5. Learn how to use combinatorics and graph theory to calculate the subgraph isomorphism, suitable for advanced high school and college students.

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