Nordhaus-gaddum type inequalities for tree covering numbers on unitary cayley graphs of finite rings

The unitary Cayley graph \Gamma_n of a finite ring \mathbb{Z}_n is the graph with vertex set \mathbb{Z}_n and two vertices x and y are adjacent if and only if x-y is a unit in \mathbb{Z}_n‎. ‎A family \mathcal{F} of mutually edge disjoint trees in \Gamma_n is called a tree cover of \Gamma_n if for each edge e\in E(\Gamma_n)‎, ‎there exists a tree T\in\mathcal{F} in which e\in E(T)‎. ‎The minimum cardinality among tree covers of \Gamma_n is called a tree covering number and denoted by \tau(\Gamma_n)‎. ‎In this paper‎, ‎we prove that‎, ‎for a positive integer n\geq ۳ ‎, ‎the tree covering number of \Gamma_n is \displaystyle\frac{\varphi(n)}{۲}+۱ and the tree covering number of \overline{\Gamma}_n is at most n-p where p is the least prime divisor of n‎. ‎Furthermore‎, ‎we introduce the Nordhaus-Gaddum type inequalities for tree covering numbers on unitary Cayley graphs of rings \mathbb{Z}_n‎.

Keywords:

‎Nordhaus-Gaddum type inequalities‎ , ‎Unitary Cayley graph‎ , ‎Tree cover‎ , ‎Tree covering number