Editing Uniquely colorable graph (section)

==Examples==

A [[complete graph]] is uniquely colorable, because the only proper coloring is one that assigns each vertex a different color.

Every [[k-tree|''k''-tree]] is uniquely (''k''&nbsp;+&nbsp;1)-colorable. The uniquely 4-colorable [[planar graph]]s are known to be exactly the [[Apollonian network]]s, that is, the planar 3-trees.{{sfnp|Fowler|1998}}

Every connected [[bipartite graph]] is uniquely 2-colorable. Its 2-coloring can be obtained by choosing a starting vertex arbitrarily, coloring the vertices at even distance from the starting vertex with one color, and coloring the vertices at odd distance from the starting vertex with the other color.{{sfnp|Mahmoodian|1998}}