Vol. 11, No. 2, 2018

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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