On Kotzigʹs conjecture concerning graphs with a unique regular path-connectivity

Kotzig (see Bondy and Murty, Graph Theory with Applications, North-Holland, Amsterdam, 1976) conjectured that there exists no graph with the property that every pair of vertices is connected by a unique path of length k, k>2. Kotzig (Graph Theory and Related Topics, Academic Press, New York, 1979, pp. 358–367) has proved this conjecture for 211. Here we prove this conjecture for the remaining cases k=9,10,11.