thumb|right|Example of Exact Coloring with 7 colors and 14 vertices
In graph theory, an exact coloring is a (proper) vertex coloring in which every pair of colors appears on exactly one pair of adjacent vertices.
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.
Complete graphs, detachments, and Euler tours
thumb|Exact coloring of the [[complete graph K<sub>6</sub>]]
Every n-vertex complete graph K<sub>n</sub> has an exact coloring with n colors, obtained by giving each vertex a distinct color.
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. However, the problem may be solved in polynomial time for trees of bounded degree.