Domination number of cubic graphs with large girth

TL;DR

This paper establishes an upper bound on the domination number of large-girth cubic graphs, showing it is less than approximately 30% of the number of vertices, with an asymptotic improvement as girth increases.

Contribution

It provides a new upper bound on the domination number for cubic graphs with large girth, improving previous estimates by a significant margin.

Findings

Domination number is less than 0.299871n + O(n/g) for large-girth cubic graphs.

As girth increases, the domination number approaches less than 3n/10.

The result improves understanding of domination in sparse, regular graphs.

Abstract

We show that every n-vertex cubic graph with girth at least g have domination number at most 0.299871n+O(n/g)<3n/10+O(n/g).