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

Abstract

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 $ein E(Gamma_n)$‎, ‎there exists a tree $Tinmathcal{F}$ in which $ein E(T)$‎. ‎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 $ ngeq 3 $‎, ‎the tree covering number of $ Gamma_n $ is $ displaystylefrac{varphi(n)}{2}+1 $ 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 $mathbb{Z}_n$‎.