| Affiliation: | a Department of Mathematics, Shijiazhuang College, Shijiazhuang 050035, PR China
b Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan
c Taida Institute for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan
d National Center for Theoretical Sciences, Taipei Office, Taiwan |
| Abstract: | Since 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. |
