Construction of uniquely vertex k-colorable graphs with minimum possible size

Abstract

It was shown that a uniquely vertex k-colorable graph of order n has minimum size n(k − 1) − (2k), and a uniquely vertex 3-colorable extremal graph with the minimum degree 3 can be constructed. In this note, we construct an infinite family of uniquely vertex k-colorable graphs of the order n, the size n(k − 1) − (2k) and the minimum degree k by using a recursion method.