AbstractA graph is called uniquely colorable if there is only one partition of its point set into the smallest possible number of color classes. Theorems concerning uniquely colorable graphs G include:o(1)The subgraph of G induced by the union of two color classes is connected.(2)Every homomorphic image of G is also uniquely colorable.(3)If a single new point w is added to G, and also at least one line joining w with each color class of G, then the resulting graph is uniquely colorable.(4)If u is a point of G of degree χ(G)−1, then G−u is uniquely colorable.A result concerning n-colorable graphs is strengthened by a proof that, for all n≥3, there exists a uniquely n-colorable graph not containing Kn.