Unique maximum independent sets in graphs on monomials of a fixed degree

TL;DR

This paper investigates the structure of graphs formed by monomials of fixed degree, proving uniqueness of maximum independent sets in specific cases and exploring domination properties, with conjectures for broader scenarios.

Contribution

It introduces a novel class of graphs based on monomials and establishes conditions for the uniqueness of maximum independent sets, advancing combinatorial understanding.

Findings

Unique maximum independent sets when n=3 and d divisible by 3

Unique maximum independent sets when n=4 and d is even

Conjecture that domination number equals independent domination number in all cases

Abstract

We consider graphs on monomials in $n$ variables of a fixed degree $d$ where two monomials are adjacent if and only if their least common multiple has degree $d+1$. We prove that when $n = 3$ and $d$ is divisible by $3$ as well as when $n=4$ and $d$ is even that these graphs have a unique maximum independent set. Domination in these graphs is also considered, and we conjecture that there is equality of the domination number and independent domination number in all cases.