Exact coloring

That is, it is a partition of the vertices of the graph into disjoint independent sets such that, for each pair of distinct independent sets in the partition, there is exactly one edge with endpoints in each set.

Every graph with an n-color exact coloring may be obtained as a detachment of a complete graph, a graph obtained from the complete graph by splitting each vertex into an independent set and reconnecting each edge incident to the vertex to exactly one of the members of the corresponding independent set.

edges has an exact coloring, obtained by forming an exact coloring of the complete graph Kk and then finding an Euler tour of this complete graph.

A graph G with n vertices and m edges has a harmonious k-coloring if and only if

and there exists a subgraph H of G with an exact k-coloring in which each edge of G − H has endpoints of different colorings.

The need for the condition on the edges of G − H is shown by the example of a four-vertex cycle, which has a subgraph with an exact 3-coloring (the three-edge path) but does not have a complete 3-coloring itself.

[1][3] However, the problem may be solved in polynomial time for trees of bounded degree.