In a school, there are $n$ students and some of them are friends each other. (Friendship is mutual.) Define $ a, b $ the minimum value which satisfies the following conditions:
(1) We can divide students into $ a $ teams such that two students in the same team are always friends.
(2) We can divide students into $ b $ teams such that two students in the same team are never friends.
Find the maximum value of $ N = a+b $ in terms of $n$.
1. **Base Case:**
   For \( n = 1 \), the graph \( G \) has only one vertex. Thus, \( \chi(G) = 1 \) and \( \chi(G^C) = 1 \). Therefore, \( \chi(G) + \chi(G^C) = 1 + 1 = 2 \).

2. **Inductive Hypothesis:**
   Assume that for any graph \( G_0 \) with \( k \) vertices, \( \chi(G_0) + \chi(G_0^C) \leq k + 1 \).

3. **Inductive Step:**
   Consider a graph \( G \) with \( k + 1 \) vertices. We need to show that \( \chi(G) + \chi(G^C) \leq k + 2 \).

   - **Case 1:**
     Suppose there exists a vertex \( v \) in \( G \) such that \( \deg(v) < \chi(G) - 1 \). Let \( G_0 \) be the graph obtained by removing \( v \) from \( G \). By the minimality of \( \chi(G) \), we have \( \chi(G_0) = \chi(G) \).

     Since \( \deg(v) < \chi(G) - 1 \), when we add \( v \) back, there is at least one color available for \( v \). Thus, \( \chi(G) \) remains the same. For the complement graph \( G^C \), adding \( v \) can increase the chromatic number by at most 1. Therefore, \( \chi(G^C) \leq \chi(G_0^C) + 1 \).

     Using the inductive hypothesis:
     \[
     \chi(G) + \chi(G^C) \leq \chi(G_0) + \chi(G_0^C) + 1 \leq (k + 1) + 1 = k + 2
     \]

   - **Case 2:**
     Suppose for every vertex \( v \) in \( G \), \( \deg(v) \geq \chi(G) - 1 \). This implies \( \Delta(G^C) \leq k + 1 - \chi(G) \).

     Using the fact that \( \chi(G) \leq \Delta(G) + 1 \) and \( \chi(G^C) \leq \Delta(G^C) + 1 \):
     \[
     \chi(G^C) \leq k + 2 - \chi(G)
     \]
     Therefore:
     \[
     \chi(G) + \chi(G^C) \leq \chi(G) + (k + 2 - \chi(G)) = k + 2
     \]

4. **Conclusion:**
   By induction, we have shown that for any graph \( G \) with \( n \) vertices, \( \chi(G) + \chi(G^C) \leq n + 1 \).

5. **Maximum Value:**
   The maximum value is achieved when \( G \) is a complete graph \( K_n \). In this case, \( \chi(G) = n \) and \( \chi(G^C) = 1 \), giving:
   \[
   \chi(G) + \chi(G^C) = n + 1
   \]

The final answer is \( \boxed{ n + 1 } \).