Domination in Kn\"odel Graphs

TL;DR

This paper investigates the domination problem in Kn"odel graphs, providing explicit bounds on the domination number using elementary number theory, especially when certain prime divisors with primitive roots exist.

Contribution

It introduces new bounds for the domination number in Kn"odel graphs leveraging elementary number theory and conditions on prime divisors with primitive roots.

Findings

Explicit upper bounds for the domination number of Kn"odel graphs.

Connection between prime divisors with primitive roots and domination properties.

Application of elementary number theory techniques to graph domination problems.

Abstract

Given a graph and an integer $k$, it is an NP-complete problem to decide whether there is a dominating set of size at most $k$. In this paper we study this problem for the Kn\"odel Graph on $n$ vertices using elementary number theory techniques. In particular, we show an explicit upper bound for the domination number of the Kn\"odel Graph on $n$ vertices any time that we can find a prime number $p$ dividing $n$ for which $2$ is a primitive root.