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In this paper, we consider the problem of classifying commutative rings according to the genus number of its associating zero-divisor graphs. The zero-divisor graph of <i>R</i>, where <i>R</i> is a commutative ring with nonzero identity, denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">Γ</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula>, is the undirected graph whose vertices are the nonzero zero-divisors of <i>R</i>, and the distinct vertices <i>x</i> and <i>y</i> are adjacent if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mi>y</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. Here, we classify the local rings with genus three zero-divisor graphs.