Note on the m-step competition numbers of paths and cycles

AbstractSince Cohen introduced the competition graph in 1968, the competition graph has been studied widely and many variations of it have been discussed. In 2000, Cho, Kim and Nam defined and studied the m-step competition graph and computed the 2-step competition numbers of paths and cycles. Recently, Ho gave bounds for the m-step competition numbers of paths and cycles. In this paper we continue to study the m-step competition numbers of paths and cycles, partially improve the results of Ho on the upper bounds of the m-step competition numbers of paths and cycles, and show that a conjecture posed by Ho is not true