Suppose that R is a commutative ring with identity. Let \(\Omega (R)\) be the set of all nonzero proper principal ideals of R, and let (x) be the principal ideal generated by \(x\in R.\) The reduced cozero-divisor graph \(\Gamma _r(R)\) of R is defined as a graph with the vertex set \(\Omega (R)\) and two distinct vertices (x) and (y) in \(\Omega (R)\) are adjacent if and only if \((x)\nsubseteq (y)\) and \((y)\nsubseteq (x).\) In this paper, we present some results on the degree, clique number and chromatic number of the reduced cozero-divisor graph of reduced rings. Also Eulerian, Hamiltonicity nature, pancyclic, perfect and weakly perfect reduced cozero-divisor graphs of reduced rings are investigated. Finally, we classify all Artin non-local rings whose reduced cozero-divisor graph has crosscap at most two.
