2019/08/07 by Vemuri, Harish · 1 citation
#05C69 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1908.02445
Let Xn denote the unitary Cayley graph of ℤ/nℤ. We continue the study of cases in which the inequality γt(Xn) ≤ g(n) is strict, where γt denotes the total domination number, and g is the arithmetic function known as Jacobsthal's function. The best that is currently known in this direction is a construction of Burcroff which gives a family of n with arbitrarily many prime factors that satisfy γt(Xn) ≤ g(n)-2. We present a new interpretation of the problem which allows us to use recent results on the computation of Jacobsthal's function to construct n with arbitrarily many prime factors that satisfy γt(Xn) ≤ g(n)-16. We also present new lower bounds on the domination numbers of direct products of complete graphs, which in turn allow us to derive new asymptotic lower bounds on γ(Xn), where γ denotes the domination number. Finally, resolving a question of Defant and Iyer, we completely classify all graphs G = ∏i=1t Kni satisfying γ(G) = t+2.