Example 1 Consider a complete graph with $n$ vertices. The vertices and edges of this complete graph are colored according to the following rules:
(1) Two edges emanating from the same vertex have different colors;
(2) The color of a vertex is different from the colors of the edges emanating from it.

For each fixed $n$, find the minimum number of colors required. ${ }^{[1]}$
(2007, Italian National Team Selection Exam)
【Analysis】Since the same vertex can lead to $n-1$ edges of different colors, and the color of this vertex is different from the colors of these $n-1$ edges, thus, at least $n$ colors are needed.
Solution The minimum value is $n$.
Let the $n$ vertices be $v_{0}, v_{2}, \cdots, v_{n-1}$, and the $n$ colors be $C_{0}, C_{1}, \cdots, C_{n-1}$. Color the edge $v_{i} v_{j}$ with color $C_{i+j} (i+j$ represents the remainder of the sum of $i$ and $j$ modulo $n)$, and color the vertex $v_{i}$ with color $C_{i+i}$. Then this coloring scheme satisfies the problem's requirements.