Finding cycles in the k-th power digraphs over the integers modulo a prime

Dresden, Greg; Tu, Wenda

doi:10.2140/involve.2018.11.181

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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

DOI: 10.2140/involve.2018.11.181

Abstract

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

Keywords

digraphs, cycles, graph theory, number theory

Mathematical Subject Classification 2010

Primary: 05C20

Secondary: 11R04

Milestones

Received: 21 January 2014

Revised: 8 June 2017

Accepted: 21 June 2017

Published: 17 September 2017

Communicated by Kenneth S. Berenhaut

Authors

Greg Dresden
Department of Mathematics Washington and Lee University Lexington, VA United States

Wenda Tu
Department of Statistics and Actuarial Science University of Iowa Iowa City, IA United States

Vol. 11, No. 2, 2018