Examples
A complete graph is uniquely colorable, because the only proper coloring is one that assigns each vertex a different color.
Every k-tree is uniquely (k + 1)-colorable. The uniquely 4-colorable planar graphs are known to be exactly the Apollonian networks, that is, the planar 3-trees (Fowler 1998).
Read more about this topic: Uniquely Colorable Graph
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