Domination in direct products of complete graphs

Graphs Defined on Rings: A Review

“…Following the notion of colouring, domination in unitary Cayley graphs was investigated in [107][108][109][110]. In [98], the domination number, upper domination number, and total domination number (refer to [34]) of the unitary Cayley graphs were investigated based on the structural property of the unitary Cayley graph X n to be realised, as a direct product of its factor graphs that were complete.…”

“…The study on the domination parameters of the unitary Cayley graph X n was extended in [110], where the open problem to find an integer n such that γ t (X n ) ≥ g(n) − 3 was solved, using the updated results on the nature of Jacobsthal function in the literature. The problem was solved not just for γ t (X n ) ≥ g(n) − 3, but also the existence of n with arbitrarily many prime factors that satisfied γ t (X n ) ≥ g(n) − 16 was also proven in [110].…”

“…The study on the domination parameters of the unitary Cayley graph X n was extended in [110], where the open problem to find an integer n such that γ t (X n ) ≥ g(n) − 3 was solved, using the updated results on the nature of Jacobsthal function in the literature. The problem was solved not just for γ t (X n ) ≥ g(n) − 3, but also the existence of n with arbitrarily many prime factors that satisfied γ t (X n ) ≥ g(n) − 16 was also proven in [110]. In addition to this, new lower bounds on the domination numbers of direct products of complete graphs were presented in [110], from which new asymptotic lower bounds on the domination number of X n , when n was a product of distinct primes, were obtained by adopting the proof techniques used in [108].…”

Unimodality and monotonic portions of certain domination polynomials

“…In general, this relationship can be rather complex, so we focus on a family of graphs that has been shown to have interesting domination properties in other contexts: direct and Cartesian products of complete graphs. The direct product of complete graphs has been shown to have extremal properties in the domination chain [9,11,16], while the Cartesian products of two complete graphs were used by the first author to demonstrate the tightness of Brešar, Klavžar, and Rall's inequality Γ(G × H) ≥ Γ(G)Γ(H), which holds for any graphs G and H [10,7].…”

“…In Sections 5.1 and 5.2, we prove the unimodality of the direct product of arbitrarily many complete graphs and the Cartesian product of two complete graphs, respectively. The former family includes the unitary Cayley graphs of Z/qZ for every squarefree integer q, which have received recent attention for their extremal domination properties [9,10,11,16].…”