Abstract. Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Γ′ (R) is a graph with vertex set W∗ (R), where W∗ (R) is the set of all nonzero and non-unit elements of R, and two distinct vertices a and b are adjacent if and only if a /∈ (b) and b /∈ (a). In this paper, we study the primitive cycle of cozero-divisor graphs and we determine all Artinian rings with claw-free or triangle-free cozero-divisor graphs. Also, we investigate all Artinian rings whose cozero-divisor graphs are C4-free

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