LOCALLY SEMICOMPLETE DIGRAPHS WITH A FACTOR COMPOSED OF k CYCLES

A digraph is locally semicomplete if for every vertex $\chi$, the set of in-neighbors as well as the set of out-neighbors of $\chi$ induce semicomplete digraphs. Let D be a k-connected locally semicomplete digraph with k $\geq$ 3 and g denote the length of a longest induced cycle of D. It is shown that if D has at least 7(k-1)g vertices, then D has a factor composed of k cycles; furthermore, if D is semicomplete and with at least 5k ＋ 1 vertices, then D has a factor composed of k cycles and one of the cycles is of length at most 5. Our results generalize those of [3] for tournaments to locally semicomplete digraphs.