1. Total resistance of the wire

The uniform wire has total resistance $R = 18\,\Omega.$

It is bent into an equilateral triangle, so the wire is divided into 3 equal sides.

Hence, resistance of each side is $R_{\text{side}} = \frac{18}{3} = 6\,\Omega.$

2. Resistance between any two vertices

Let the vertices be $A, B, C$. We need the equivalent resistance between two vertices, say $A$ and $B$.

There are two possible paths from $A$ to $B$:

• Direct side $AB$: resistance $= 6\,\Omega$
• Path through the third vertex $C$: resistance $= AC + CB = 6 + 6 = 12\,\Omega$

So between $A$ and $B$, we have two resistances in parallel: $6\,\Omega \quad \text{and} \quad 12\,\Omega.$

3. Equivalent resistance

Using parallel combination, $R_{eq} = \frac{6 \times 12}{6 + 12} = \frac{72}{18} = 4\,\Omega.$

4. Final answer

Therefore, the resistance between any two vertices of the triangle is $\boxed{4\,\Omega}.$

So the correct option is C.