Roman Domination of Cartesian Bundles of Cycles over Cycles - Simon Brezovnik; Janez Žerovnik (Mathematics)
A Roman dominating function f of a graph <semantics>G=(V,E)</semantics> assigns labels from the set <semantics>{0,1,2}</semantics> to vertices such that every vertex labeled 0 has a neighbor labeled 2. The weight of an RDF f is defined as <semantics>w(f)=∑v∈Vf(v)</semantics>, and the Roman domination number, <semantics>γR(G)</semantics>, is the minimum weight among all RDFs of G. This paper studies the domination and Roman domination numbers in Cartesian bundles of cycles. Furthermore, the constructed optimal patterns improve known bounds and suggest even better bounds might be achieved by combining patterns, especially for bundles involving shifts of order <semantics>4k</semantics> and <semantics>5k</semantics>.