Domination and upper domination of direct product graphs

Abstract

Let X Z/nZ denote the unitary Cayley graph of Z/nZ. We present results on the tightness of the known inequality γ(X Z/nZ ) ≤ γt(X Z/nZ ) ≤ g(n), where γ and γt denote the domination number and total domination number, respectively, and g is the arithmetic function known as Jacobsthal's function. In particular, we construct integers n with arbitrarily many distinct prime factors such that γ(X Z/nZ ) ≤ γt(X Z/nZ ) ≤ g(n) − 1. We give lower bounds for the domination numbers of direct products of complete graphs and present a conjecture for the exact values of the upper domination numbers of direct products of balanced, complete multipartite graphs.

Domination parameters of the unitary Cayley graph of \mathbb{Z}/n\mathbb{Z}

“…In a preprint of the current article, we conjectured that the upper irredundance number was equal to the independence number for all direct products of balanced, complete multipartite graphs. This was a strengthening of an earlier conjecture of Defant and Iyer [11,Conjecture 4.2]. Since the online release of that preprint, Alon and Defant [4, Theorem 1.2] have confirmed the stronger conjecture for all but 37 such products, thus improving upon Corollary 5.6.…”

“…We provide an upper bound on the lower independence number of X n in Corollary 4.12, thus demonstrating an error in a result of Uma Maheswari and Maheswari [20,Theorem 4.2]. Defant and Iyer [11] recently determined the value of γ 4 i=1 K n i in several cases; we compute this parameter in all cases. In Section 5, we provide upper bounds on the sizes of irredundant sets in direct products of balanced, complete multipartite graphs.…”

“…In this paper, we build upon the work of Defant and Iyer [11] to determine domination parameters of the unitary Cayley graphs of Z/nZ. Let g(n) denote the minimum positive integer m such that every set of m consecutive integers contains an integer which is coprime to n; this arithmetic function is known as the Jacobsthal function.…”

Isoperimetry, stability, and irredundance in direct products

“…It is straightforward to check that α(G) = |V (G)|/t n (alternatively, β(G) = 1/t n ). While studying domination parameters of unitary Cayley graphs, the second author and Iyer were led to conjecture that for these graphs α(G) = Γ(G) [20]. They proved this conjecture in the cases t n ≤ 2 and n ≤ 3.…”

“…Invoking (20), we obtain the inequalities (22) µ((I \ J 1,j ) ∩ J 1,n ) = µ(I \ J 1,j ) − µ(I \ (J 1,j ∪ J 1,n )) ≥ (2t j − 1)(t j − 1) t 4 j − t n − 1 t n · t j − 1 t 5 j . and (23) µ((I \ J 1,n ) ∩ J 1,j ) = µ(I \ J 1,n ) − µ(I \ (J 1,j ∪ J 1,n )) ≥ (2t n − 1)(t n − 1) t 4…”