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 γn 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.