Domination and maximal intersecting families in comaximal graphs of commutative rings

Abstract

The comaximal graph $\mathcal{G}(R)$ of a commutative ring $R$ has as vertices the nonunits outside the Jacobson radical, two being adjacent if they generate $R$ as an ideal. For rings with $k\ge 4$ maximal ideals, only the bounds $2\leγ(\mathcal{G}(R))\le k$ were known, to our knowledge. We prove $γ=γ_t=γ_c=k$ whenever $3\le k<\infty$, without finiteness or Artinian hypotheses, classify the minimum dominating sets, and determine the domatic number. Independent dominating sets correspond to maximal intersecting families of subsets of $\{1,\dots,k\}$; we obtain a closed formula for the independent domination number of every finite non-local ring, which can be exponentially larger than the domination number. The proofs rest on the class sizes of the blow-up description of $\mathcal{G}(R)$ due to Moconja and Petrović, which we determine for every ring with finitely many maximal ideals, finite or infinite; with them, domination problems on $\mathcal{G}(R)$ become weighted extremal problems on subsets of $\{1,\dots,k\}$, and the formula for the independent domination number follows from the multiplicative form of these sizes. We also treat rings with infinitely many maximal ideals, graph polynomials, and lattices whose maximal ideals are prime.