2493 - Divide Nodes Into the Maximum Number of Groups\.

Hard

You are given a positive integer n representing the number of nodes in an undirected graph. The nodes are labeled from 1 to n.

You are also given a 2D integer array edges, where <code>edgesi = a_i, b_i</code> indicates that there is a bidirectional edge between nodes <code>a_i</code> and <code>b_i</code>. Notice that the given graph may be disconnected.

Divide the nodes of the graph into m groups ( 1-indexed ) such that:

• Each node in the graph belongs to exactly one group.

• For every pair of nodes in the graph that are connected by an edge <code>a_i, b_i</code>, if <code>a_i</code> belongs to the group with index x, and <code>b_i</code> belongs to the group with index y, then |y - x| = 1.

Return the maximum number of groups (i.e., maximum m) into which you can divide the nodes. Return -1 if it is impossible to group the nodes with the given conditions.

Example 1:

Input: n = 6, edges = \[\[1,2],1,4,1,5,2,6,2,3,4,6]

Output: 4

Explanation: As shown in the image we:

• Add node 5 to the first group.

• Add node 1 to the second group.

• Add nodes 2 and 4 to the third group.

• Add nodes 3 and 6 to the fourth group.

We can see that every edge is satisfied. It can be shown that that if we create a fifth group and move any node from the third or fourth group to it, at least on of the edges will not be satisfied.

Example 2:

Input: n = 3, edges = \[\[1,2],2,3,3,1]

Output: -1

Explanation: If we add node 1 to the first group, node 2 to the second group, and node 3 to the third group to satisfy the first two edges, we can see that the third edge will not be satisfied. It can be shown that no grouping is possible.

Constraints:

• 1 <= n <= 500

• <code>1 <= edges.length <= 10⁴</code>

• edges[i].length == 2

• <code>1 <= a_i, b_i<= n</code>

• <code>a_i != b_i</code>

• There is at most one edge between any pair of vertices.