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

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.