Color tree with minimum colors such that colors of edges incident to a vertex are different

Given a tree with N nodes. The task is to color the tree with the minimum number of colors(K) such that the colors of the edges incident to a vertex are different. Print K in first-line and then in next line print N – 1 space-separated integer represents the colors of the edges.

Examples:

Input: N = 3, edges[][] = {{0, 1}, {1, 2}}
                   0
                  /  
                 1
                /
               2  
Output:
2
1 2
                   0
                  / (1) 
                 1
                / (2)
               2

Input: N = 8, edges[][] = {{0, 1}, {1, 2}, 
                           {1, 3}, {1, 4}, 
                           {3, 6}, {4, 5}, 
                           {5, 7}}
                    0
                   /
                  1
                / \ \
               2   3 4
                  /   \
                 6     5
                        \
                         7
Output:
4
1 2 3 4 1 1 2

Approach: First, let’s think about the minimum number of colors K. For every vertex v, deg(v) ≤ K should meet (where deg(v) denotes the degree of vertex v). In fact, there exists a vertex with all K different colors. First, choose a vertex and let it be the root, thus T will be a rooted tree. Perform a breadth-first search from the root. For each vertex, determine the colors of edges between its children one by one. For each edge, use the color with the minimum index among those which are not used as colors of edges whose one of endpoints is the current vertex. Then each index of color does not exceed K.

Below is the implementation of the above approach:

CPP

// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;

// Add an edge between the vertexes
void add_edge(vector<vector<int> >& gr, int x,
              int y, vector<pair<int, int> >& edges)
{
    gr[x].push_back(y);
    gr[y].push_back(x);
    edges.push_back({ x, y });
}

// Function to color the tree with minimum
// number of colors such that the colors of
// the edges incident to a vertex are different
int color_tree(int n, vector<vector<int> >& gr,
               vector<pair<int, int> >& edges)
{

    // To store the minimum colors
    int K = 0;

    // To store color of the edges
    map<pair<int, int>, int> color;

    // Color of edge between its parent
    vector<int> cs(n, 0);

    // To check if the vertex is
    // visited or not
    vector<int> used(n, 0);

    queue<int> que;
    used[0] = 1;
    que.emplace(0);

    while (!que.empty()) {

        // Take first element of the queue
        int v = que.front();
        que.pop();

        // Take the possible value of K
        if (K < (int)gr[v].size())
            K = gr[v].size();

        // Current color
        int cur = 1;

        for (int u : gr[v]) {

            // If vertex is already visited
            if (used[u])
                continue;

            // If the color is similar
            // to it's parent
            if (cur == cs[v])
                cur++;

            // Assign the color
            cs[u] = color[make_pair(u, v)]
                = color[make_pair(v, u)] = cur++;

            // Mark it visited
            used[u] = 1;

            // Push into the queue
            que.emplace(u);
        }
    }

    // Print the minimum required colors
    cout << K << endl;

    // Print the edge colors
    for (auto p : edges)
        cout << color[p] << " ";
}

// Driver code
int main()
{
    int n = 8;

    vector<vector<int> > gr(n);
    vector<pair<int, int> > edges;

    // Add edges
    add_edge(gr, 0, 1, edges);
    add_edge(gr, 1, 2, edges);
    add_edge(gr, 1, 3, edges);
    add_edge(gr, 1, 4, edges);
    add_edge(gr, 3, 6, edges);
    add_edge(gr, 4, 5, edges);
    add_edge(gr, 5, 7, edges);

    // Function call
    color_tree(n, gr, edges);

    return 0;
}

Python

# Python3 implementation of the approach
from collections import deque as queue

gr = [[] for i in range(100)]
edges = []

# Add an edge between the vertexes
def add_edge(x, y):
    gr[x].append(y)
    gr[y].append(x)
    edges.append([x, y])

# Function to color the tree with minimum
# number of colors such that the colors of
# the edges incident to a vertex are different
def color_tree(n):

    # To store the minimum colors
    K = 0

    # To store color of the edges
    color = dict()

    # Color of edge between its parent
    cs = [0] * (n)

    # To check if the vertex is
    # visited or not
    used = [0] * (n)

    que = queue()
    used[0] = 1
    que.append(0)

    while (len(que) > 0):

        # Take first element of the queue
        v = que.popleft()

        # Take the possible value of K
        if (K < len(gr[v])):
            K = len(gr[v])

        # Current color
        cur = 1

        for u in gr[v]:

            # If vertex is already visited
            if (used[u]):
                continue

            # If the color is similar
            # to it's parent
            if (cur == cs[v]):
                cur += 1

            # Assign the color
            cs[u] = cur
            color[(u, v)] = color[(v, u)] = cur
            cur += 1

            # Mark it visited
            used[u] = 1

            # Push into the queue
            que.append(u)

    # Print minimum required colors
    print(K)

    # Print edge colors
    for p in edges:
        i = (p[0], p[1])
        print(color[i], end = " ")

# Driver code
n = 8

# Add edges
add_edge(0, 1)
add_edge(1, 2)
add_edge(1, 3)
add_edge(1, 4)
add_edge(3, 6)
add_edge(4, 5)
add_edge(5, 7)

# Function call
color_tree(n)

# This code is contributed by mohit kumar 29

Output:

4
1 2 3 4 1 1 2