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The perfect matching condition and forbidden even cycles subgraphs are essential in finding large bisections of graphs. In this paper, we show that the perfect matching condition can be replaced by the minimum degree condition. Let $C_{\\ell}$ be a cycle of length $\\ell$ for $\\ell\\ge 3$, and let $G$ be a $\\{C_4, C_6\\}$-free graph with $m$ edges and minimum degree at least 2. 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