The Existence of LHMTS(mv) and LHDTS(mv)

In this article, we establish the existence of an LHMTS(mv) for v . 2 (mod 6) and m . 3 (mod 6). Thus there exists an LHMTS(mv) if and only if v(v . 1)m2 . 0 (mod 3) except possibly for v = 6, m . 1,5 (mod 6) and m . 1. In the similar way, the existence of LHDTS(mv) is completely determined, i.e., there exists an LHDTS(mv) if and only if v(v . 1)m2 . 0 (mod 3).