The 2-rainbow domination number of Cartesian bundles over cycles

A k-rainbow dominating function (kRDF) f of G assigns subsets of \(\{1,2,\ldots ,k\}\) $\{1,2,\dots ,k\}$ to vertices, such that for vertex v with \(f(v)=\emptyset\) $f(v)=\mathrm{\varnothing }$ , \(\bigcup \nolimits _{u\in N(v)}f(u)=\{1,2,\ldots ,k\}\) ${\bigcup }_{u\in N(v)}f(u)=\{1,2,\dots ,k\}$ . The weight w(f) of kRDF f is \(w(f)=\sum _{v\in V(G)}\left| f(v)\right|\) $w(f)={\sum }_{v\in V(G)}\left|f,(,v,)\right|$ . The minimum weight of a kRDF of G is the k-rainbow domination number denoted by \(\gamma _{rk}(G)\) ${\gamma }_{\mathit{rk}}(G)$ . This paper focuses on the 2-rainbow domination number of Cartesian graph bundles of cycles over cycles, extending recent results for Cartesian product of cycles. Exact values are given for certain infinite families, and tight lower and upper bounds are established for general case.