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Graph with only one possible coloring

In graph theory, a uniquely colorable graph is a k chromatic graph that has only one possible (proper) k coloring up to permutation of the colors. Equivalently, there is only one way to partition its vertices into k independent sets and there is no way to partition them into k − 1 independent sets.

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. 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.

A minimal imperfect graph is a graph in which every subgraph is perfect. The deletion of any vertex from a minimal imperfect graph leaves a uniquely colorable subgraph.

A uniquely edge colorable graph is a k edge chromatic graph that has only one possible (proper) k edge coloring up to permutation of the colors. The only uniquely 2 edge colorable graphs are the paths and the cycles. For any k, the stars K1,k are uniquely k edge colorable. Moreover, conjectured and proved that, when k ≥ 4, they are also the only members in this family. However, there exist uniquely 3 edge colorable graphs that do not fit into this classification, such as the graph of the triangular pyramid. If a cubic graph is uniquely 3 edge colorable, it must have exactly three Hamiltonian cycles, formed by the edges with two of its three colors, but some cubic graphs with only three Hamiltonian cycles are not uniquely 3 edge colorable. Every simple planar cubic graph that is uniquely 3 edge colorable contains a triangle, but observed that the generalized Petersen graph G(9,2) is non planar, triangle free, and uniquely 3 edge colorable. For many years it was the only known such graph, and it had been conjectured to be the only such graph but now infinitely many triangle free non planar cubic uniquely 3 edge colorable graphs are known.