Uniquely colorable graphs

A graph is called uniquely k-colorable if there is only one partition of its vertex set into k color classes. The first result of this note is that if a k-colorable graph G of order n is such that its minimal degree, δ(G), is greater than (3k−5)/(3k−2) n then it is uniquely k-colorable. This result can be strengthened considerably if one considers only graphs having an obvious property of k-colorable graphs. More precisely, the main result of the note states the following. If G is a graph of order n that has a k-coloring in which the subgraph induced by the union of any two color classes is connected then δ(G)>(1−(1/(k−1))) n implies that G is uniquely k-colorable. Both these results are best possible