3. 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 each vertex is different from the colors of the edges emanating from it.

For each fixed $n$, find the minimum number of colors required.
Obviously, at least $n$ colors are needed, because the $n-1$ edges extending from a single point must all have different colors, and they must also be different from the color of that point.

A coloring scheme that meets the conditions of the problem using exactly $n$ colors does exist.

Let the $n$ points be $v_{0}, v_{1}, \cdots, v_{n-1}$, and the $n$ colors be $C_{0}, C_{1}, \cdots, C_{n-1}$.

Color the edge $v_{i} v_{j}$ with color $C_{i+j}$, and color the point $v_{i}$ with color $C_{i+i}$. Then this coloring scheme satisfies the problem's requirements.