An island has $10$ cities, where some of the possible pairs of cities are connected by roads. A [i]tour route[/i] is a route starting from a city, passing exactly eight out of the other nine cities exactly once each, and returning to the starting city. (In other words, it is a loop that passes only nine cities instead of all ten cities.) For each city, there exists a tour route that doesn't pass the given city. Find the minimum number of roads on the island.
1. **Understanding the Problem:**
   - We have 10 cities, and some pairs of cities are connected by roads.
   - A tour route is a loop that starts from a city, passes exactly 8 out of the other 9 cities exactly once each, and returns to the starting city.
   - For each city, there exists a tour route that doesn't pass through the given city.
   - We need to find the minimum number of roads on the island.

2. **Graph Representation:**
   - Represent the cities as vertices of a graph \( G \) with 10 vertices.
   - The roads are the edges of the graph.
   - A tour route corresponds to a 9-cycle in the graph that excludes one specific vertex.

3. **Degree Condition:**
   - For each vertex \( v \), there must be a 9-cycle that excludes \( v \).
   - If a vertex \( v \) has a degree less than 3, it cannot be part of a 9-cycle excluding \( v \) because it would not have enough connections to form such a cycle.
   - Therefore, each vertex must have a degree of at least 3.

4. **Minimum Number of Edges:**
   - Let \( d(v) \) denote the degree of vertex \( v \).
   - By the Handshaking Lemma, the sum of the degrees of all vertices is twice the number of edges: \(\sum_{v \in V} d(v) = 2|E|\).
   - Since each vertex has a degree of at least 3, we have \(\sum_{v \in V} d(v) \geq 3 \times 10 = 30\).
   - Thus, \(2|E| \geq 30\), which implies \(|E| \geq 15\).

5. **Construction of a Graph with 15 Edges:**
   - We need to construct a graph with 10 vertices and 15 edges where each vertex has a degree of exactly 3.
   - One possible construction is to use a 10-cycle and add 5 additional edges to ensure each vertex has a degree of 3.

6. **Verification of the Construction:**
   - Consider the vertices labeled \( v_1, v_2, \ldots, v_{10} \).
   - Connect the vertices in a cycle: \( v_1 - v_2 - v_3 - \ldots - v_{10} - v_1 \).
   - Add the following edges: \( v_1 - v_6, v_2 - v_7, v_3 - v_8, v_4 - v_9, v_5 - v_{10} \).
   - Each vertex now has a degree of 3, and the total number of edges is 15.

7. **Ensuring the Existence of 9-Cycles:**
   - For each vertex \( v_i \), removing \( v_i \) leaves a subgraph with 9 vertices and 14 edges.
   - This subgraph is connected and has enough edges to form a 9-cycle, ensuring that the condition of the problem is satisfied.

\[
\boxed{15}
\]