#### Vol. 11, No. 2, 2018

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Ethics Statement Editorial Login Author Index Coming Soon Contacts ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print) Other MSP Journals
Finding cycles in the $k$-th power digraphs over the integers modulo a prime

### Greg Dresden and Wenda Tu

Vol. 11 (2018), No. 2, 181–194
##### Abstract

For $p$ prime and $k\ge 2$, let us define ${G}_{p}^{\left(k\right)}$ to be the digraph whose set of vertices is $\left\{0,1,2,\dots ,p-1\right\}$ such that there is a directed edge from a vertex $a$ to a vertex $b$ if ${a}^{k}\equiv b\phantom{\rule{0.2em}{0ex}}mod\phantom{\rule{0.2em}{0ex}}p$. We find a new way to decide if there is a cycle of a given length in a given graph ${G}_{p}^{\left(k\right)}$.

##### Keywords
digraphs, cycles, graph theory, number theory
Primary: 05C20
Secondary: 11R04