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.

v2 has minor typographical corrections, shortened version accepted to proceedings of XI LAGOS 2021