## Coloured Graphs

### Problem 677

Let $g(n)$ be the number of **undirected graphs** with $n$ nodes satisfying the following properties:

- The graph is connected and has no cycles or multiple edges.
- Each node is either red, blue, or yellow.
- A red node may have no more than 4 edges connected to it.
- A blue or yellow node may have no more than 3 edges connected to it.
- An edge may not directly connect a yellow node to a yellow node.

For example, $g(2)=5$, $g(3)=15$, and $g(4) = 57$.

You are also given that $g(10) = 710249$ and $g(100) \equiv 919747298 \pmod{1\,000\,000\,007}$.

Find $g(10\,000) \bmod 1\,000\,000\,007$.