Regular independent sets

TL;DR

This paper investigates the regular independence number, focusing on lower bounds for various graph classes, and extends these results to the regular k-independence number for trees, forests, planar graphs, and other graph types.

Contribution

It generalizes previous bounds on the regular independence number to the regular k-independence number across multiple graph classes.

Findings

Established lower bounds for the regular k-independence number.

Extended bounds to trees, forests, planar graphs, k-trees, and k-degenerate graphs.

Provided new theoretical insights into graph invariants.

Abstract

The regular independence number, introduced by Albertson and Boutin in 1990, is the size of a largest set of independent vertices with the same degree. Lower bounds were proven for this invariant, in terms of the order, for trees and planar graphs. In this article, we generalize and extend these results to find lower bounds for the regular $k$-independence number for trees, forests, planar graphs, $k$-trees and $k$-degenerate graphs.