Domination number of graphs with minimum degree five

TL;DR

This paper establishes an upper bound of n/3 for the domination number in graphs with minimum degree five, using a novel combination of algorithmic and discharging techniques, and also offers a shorter proof for degree four graphs.

Contribution

It introduces a new proof method for bounding domination numbers in graphs with minimum degree five, improving understanding of graph domination properties.

Findings

Domination number of graphs with minimum degree five is at most n/3.

Provides a shorter proof for the degree four case with an upper bound of 4n/11.

Combines algorithmic and discharging methods for graph theoretical proofs.

Abstract

We prove that for every graph $G$ on $n$ vertices and with minimum degree five, the domination number $\gamma(G)$ cannot exceed $n/3$. The proof combines an algorithmic approach and the discharging method. Using the same technique, we provide a shorter proof for the known upper bound $4n/11$ on the domination number of graphs of minimum degree four.