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

• Difficulty Level : Expert
• Last Updated : 20 Dec, 2019

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
```

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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 ``using` `namespace` `std;`` ` `// Add an edge between the vertexes``void` `add_edge(vector >& gr, ``int` `x,``              ``int` `y, vector >& 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 >& gr,``               ``vector >& edges)``{`` ` `    ``// To store the minimum colors``    ``int` `K = 0;`` ` `    ``// To store color of the edges``    ``map, ``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 > gr(n);``    ``vector > 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
```

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