Kr-Free Uniquely Vertex Colorable Graphs with Minimum Possible Edges

There is a conjecture due to Shaoji 3], about uniquely vertex r-colorable graphs which states: \ If G is a uniquely vertex r-colorable graph with order n and size (r ? 1)n ? ? r 2 , then G contains a K r as its subgraph." In this paper for any natural number r we construct a K r-free, uniquely r-colorable graph with (r ? 1)n ? ? r 2 edges. These families of graphs are indeed counter examples to Shaoji's Conjecture. The uniquely colorable graphs are one of the main topics in graph theory. This kind of graphs have been studied extensively by diierent authors and many papers have been written about them, see say, 1], 2], 3]. Harary and his co-authors 2], mentioned that one may think that every uniquely r-colorable graph contains a subgraph isomorphic to K r. Thus in one of their theorems they proved that: \ For all r 3, there is a uniquely r-colorable graph which contains no subgraph isomorphic to K r ." Several years later, in 1990 Shaoji proved that \ If G is a uniquely r-colorable graph with order n and size m; then m (r ? 1)n ? r 2 , and the bound is the best possible" and then conjectured that \ If G is a uniquely r-colorable graph with order 1