Let n be a positive integer. Consider the list of all graphs with n vertices up to isomorphism1
.
Are there more connected or more disconnected graphs on this list? (Or perhaps there’s an equal
number?)
Hint. In this exercise, you may use without proof the following intuitively obvious fact about
disconnected graphs, which will be covered soon in lectures: a disconnected graph splits as the
union of two disjoint graphs. Formally, if G is disconnected, there exist sets A and B such that
A ∪ B = V (G), A ∩ B = ∅, and A, B ̸= ∅, such that uv /∈ E(G), for all u ∈ A and v ∈ B.

We are tasked with determining whether there are more connected or disconnected graphs on a list of all graphs with \( n \) vertices, up to isomorphism.

### Key Insights:
- **Graph**: A graph consists of a set of vertices and edges connecting some pairs of vertices.
- **Isomorphism**: Two graphs are considered isomorphic if they can be transformed into each other by relabeling the vertices.
- **Connected vs. Disconnected Graphs**:
  - A **connected graph** is one where there is a path between any two vertices.
  - A **disconnected graph** has at least two components (subgraphs) that are not connected to each other.

### Approach:

1. **Total Number of Graphs**:
   - For \( n \) vertices, the total number of graphs (without considering isomorphism) is \( 2^{\binom{n}{2}} \). This is because for each of the \( \binom{n}{2} \) pairs of vertices, we can either have an edge or no edge, giving us \( 2 \) possibilities for each pair.

2. **Counting Disconnected Graphs**:
   - A **disconnected graph** can be thought of as two or more smaller graphs on disjoint subsets of the vertex set.
   - As mentioned in the hint, if a graph is disconnected, it can be split into two disjoint sets of vertices, \( A \) and \( B \), where no edge exists between any vertex in \( A \) and any vertex in \( B \).
   - For every disconnected graph, it is possible to identify a non-empty subset \( A \subseteq V(G) \) such that there are no edges between \( A \) and its complement \( B = V(G) \setminus A \). This split gives a disconnected structure.

3. **Counting Connected Graphs**:
   - The number of **connected graphs** is the complement of the number

Let n be a positive integer. Consider the list of all graphs with n vertices up to isomorphism. Are there more connected or more disconnected graphs on this list? (Or perhaps there’s an equal number?)

Explore the comparison between connected and disconnected graphs with n vertices up to isomorphism. This problem delves into graph theory, using combinatorial formulas to count the total number of graphs, connected graphs, and disconnected graphs. Suitable for undergraduate students studying discrete mathematics.

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