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

The unitary Cayley graph Γn of a finite ring Zn is the graph with vertex set Zn and two vertices x and y are adjacent if and only if x−y is a unit in Zn. A family F of mutually edge disjoint trees in Γn is called a tree cover of Γn if for each edge e∈E(Γn), there exists a tree T∈F in which e∈E(T). The minimum cardinality among tree covers of Γn is called a tree covering number and denoted by τ(Γn). In this paper, we prove that, for a positive integer n≥3, the tree covering number of Γn is φ(n)2+1 and the tree covering number of Γ¯¯¯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 Zn.