Boxicity and Threshold Dimension of Zero Divisor Graphs
2026-08-27 • Discrete Mathematics
Discrete Mathematics
AI summaryⓘ
The authors study a special kind of graph made from a finite commutative ring, where the nodes represent elements that multiply to zero with some other element. They focus on two types of rings and figure out specific properties called boxicity and threshold dimension of these graphs. To do this, they create a new tool called the integral covering graph that helps understand the common structure of these rings. Their work also answers questions posed by other researchers about these graph properties.
zero divisor graphfinite commutative ringboxicitythreshold dimensionreduced ringsprincipal ideal domainintegral covering graphdisjointness graphgraph theory
Authors
Marco Caoduro, Meike Neuwohner
Abstract
The zero divisor graph $Γ(R)$ of a finite commutative ring $R$ has as vertices the non-zero zero divisors of $R$, with an edge between two elements exactly when their product is zero. We determine the boxicity and threshold dimension of $Γ(R)$ for two classes of finite commutative rings: reduced rings and quotients of principal ideal domains. Our proofs use a new combinatorial gadget, the integral covering graph, that captures the structure shared by both ring families and generalizes the disjointness graph on subsets of $[n]$, where two subsets are adjacent if and only if they are disjoint. In doing so, we answer two questions recently posed by L.~Sunil Chandran and Suraj Kumar Sahoo in Boxicity of Zero Divisor Graphs, Discrete Applied Mathematics 391 (2026).