How to find Time complexity of Graph coloring using backtracking?

Question

I have to find out the time complexity of graph coloring problem using backtracking. I have found somewhere it is O(n*m^n) where n=no vertex and m= number of color.

Suppose my code is given below how to find time complexity?

bool isSafe (int v, bool graph[V][V], int color[], int c)
{
    for (int i = 0; i < V; i++)
        if (graph[v][i] && c == color[i])
            return false;
    return true;
}

bool graphColoringUtil(bool graph[V][V], int m, int color[], int v)
{
    if (v == V)
        return true;

    for (int c = 1; c <= m; c++)
    {
        if (isSafe(v, graph, color, c))
        {
           color[v] = c;

           if (graphColoringUtil (graph, m, color, v+1) == true)
             return true;

           color[v] = 0;
        }
    }

    return false;
}
bool graphColoring(bool graph[V][V], int m)
{
    int *color = new int[V];
    for (int i = 0; i < V; i++)
       color[i] = 0;

    if (graphColoringUtil(graph, m, color, 0) == false)
    {
      printf("Solution does not exist\n");
      return false;
    }

    printSolution(color);
    return true;
}
void printSolution(int color[])
{
    printf("Solution Exists:"
            " Following are the assigned colors \n");
    for (int i = 0; i < V; i++)
      printf(" %d ", color[i]);
    printf("\n");
} 

Answer 1

The graphutil method will execute n times itself.It is in the c Loop,and c goes upto m . Now the c loop goes n times due to recursion(i.e. m^n) and recursion goes n times,So total it will be O(nm^n )