Given positive integers $n,k$, $n \ge 2$. Find the minimum constant $c$ satisfies the following assertion:
For any positive integer $m$ and a $kn$-regular graph $G$ with $m$ vertices, one could color the vertices of $G$ with $n$ different colors, such that the number of monochrome edges is at most $cm$.
To find the minimum constant \( c \) that satisfies the given assertion, we need to analyze the problem step-by-step.

1. **Understanding the Problem:**
   - We are given a \( kn \)-regular graph \( G \) with \( m \) vertices.
   - We need to color the vertices of \( G \) with \( n \) different colors.
   - The goal is to minimize the number of monochromatic edges (edges where both endpoints have the same color).

2. **Lower Bound:**
   - Consider the complete graph \( K_{nk+1} \) with \( nk+1 \) vertices.
   - If we color this graph with \( n \) colors, each color is used \( k \) times except one color which is used \( k+1 \) times.
   - The number of monochromatic edges in this case is at least \( n \binom{k}{2} + k \).

3. **Upper Bound:**
   - The graph \( G \) has \( \frac{mkn}{2} \) edges since it is \( kn \)-regular.
   - The chromatic number of \( G \) is at most \( kn+1 \).
   - We partition the vertices into \( nk+1 \) classes \( A_1, A_2, \ldots, A_{nk+1} \).
   - Let \( d_{ij} \) be the number of edges between \( A_i \) and \( A_j \), and \(\sum_{i<j} d_{ij} = \frac{mkn}{2} \).

4. **Coloring Strategy:**
   - Color each class \( A_i \) with one of \( n \) colors \( c_1, c_2, \ldots, c_n \).
   - Each color is used exactly \( k \) times except one color which is used \( k+1 \) times.

5. **Probability Calculation:**
   - The probability that \( A_i \) and \( A_j \) have the same color is:
     \[
     P(A_i = A_j) = \sum_{k=1}^n P(A_i = c_k, A_j = c_k) = \sum_{k=1}^n P(A_i = c_k \mid A_j = c_k) P(A_j = c_k)
     \]
   - This simplifies to:
     \[
     (n-1) \frac{k(k-1)}{(kn+1)kn} + \frac{(k+1)k}{(kn+1)kn} = \frac{nk(k-1) + 2k}{(kn+1)kn}
     \]

6. **Expectation of Monochromatic Edges:**
   - The expected number of monochromatic edges is:
     \[
     \mathbb{E} \left[ \sum \mathbb{I}_{[A_i = A_j]} d_{ij} \right] = \frac{nk(k-1) + 2k}{(kn+1)kn} \sum d_{ij} = m \frac{n \binom{k}{2} + k}{nk+1}
     \]

Therefore, the minimum constant \( c \) that satisfies the assertion is:
\[
c = \frac{n \binom{k}{2} + k}{nk+1}
\]

The final answer is \(\boxed{\frac{n \binom{k}{2} + k}{nk+1}}\).