Graph Coloring Problem

You are given an undirected graph and an integer k. Determine whether it is possible to assign one of k colors to every node so that no two nodes connected by an edge share the same color.

Input Description:

• An integer n — the number of nodes.
• An integer k — the number of available colors.
• A list of edges, each represented as a pair (u, v) of distinct integers from 0 to n-1.

Output Description:

• Return True if a valid k-coloring exists; otherwise, return False.

Example:

Input: n = 3, k = 3, edges = [[0, 1], [1, 2], [2, 0]]
Output: True

Explanation: The graph is a triangle. With three colors available, we can give each node a distinct color, satisfying the constraint.

Constraints:

• 1 <= n <= 1000
• 0 <= k <= n
• The number of edges is on the order of n