Uniquely colorable graphs

Abstract

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 (3 k −5)/(3 k −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.