D. Complete Tripartite
brute force; constructive algorithms; graphs; hashing; implementation
You have a simple undirected graph consisting of \(n\) vertices and \(m\) edges. The graph doesn't contain self-loops, there is at most one edge between a pair of vertices. The given graph can be disconnected.Let's make a definition.Let \(v_1\) and \(v_2\) be two some nonempty subsets of vertices that do not intersect. Let \(f(v_{1}, v_{2})\) be true if and only if all the conditions are satisfied: There are no edges with both endpoints in vertex set \(v_1\). There are no edges with both endpoints in vertex set \(v_2\). For every two vertices \(x\) and \(y\) such that \(x\) is in \(v_1\) and \(y\) is in \(v_2\), there is an edge between \(x\) and \(y\). Create three vertex sets (\(v_{1}\), \(v_{2}\), \(v_{3}\)) which satisfy the conditions below; All vertex sets should not be empty. Each vertex should be assigned to only one vertex set. \(f(v_{1}, v_{2})\), \(f(v_{2}, v_{3})\), \(f(v_{3}, v_{1})\) are all true. Is it possible to create such three vertex sets? If it's possible, print matching vertex set for each vertex.
The first line contains two integers \(n\) and \(m\) (\(3 \le n \le 10^{5}\), \(0 \le m \le \text{min}(3 \cdot 10^{5}, \frac{n(n-1)}{2})\)) — the number of vertices and edges in the graph.The \(i\)-th of the next \(m\) lines contains two integers \(a_{i}\) and \(b_{i}\) (\(1 \le a_{i} \lt b_{i} \le n\)) — it means there is an edge between \(a_{i}\) and \(b_{i}\). The graph doesn't contain self-loops, there is at most one edge between a pair of vertices. The given graph can be disconnected.
If the answer exists, print \(n\) integers. \(i\)-th integer means the vertex set number (from \(1\) to \(3\)) of \(i\)-th vertex. Otherwise, print \(-1\).If there are multiple answers, print any.
In the first example, if \(v_{1} = \{ 1 \}\), \(v_{2} = \{ 2, 3 \}\), and \(v_{3} = \{ 4, 5, 6 \}\) then vertex sets will satisfy all conditions. But you can assign vertices to vertex sets in a different way; Other answers like ""2 3 3 1 1 1"" will be accepted as well. In the second example, it's impossible to make such vertex sets.