On uniquely 3-colorable graphs

Abstract We show the following. (1) For each integer n ⩾12, there exists a uniquely 3-colorable graph with n vertices and without any triangles. (2) There exist infinitely many uniquely 3-colorable regular graphs without any triangles. It follows that there exist infinitely many uniquely k -colorable regular graphs having no subgraph isomorphic to the complete graph K k with k vertices for any integer k ⩾3.