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 Kk with k vertices for any integer k≥3.
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics