Every C_{2k}-free graph with minimum degree at least k has a balanced bipartition with at least m/2 + Omega(m^{(2k+1)/(2k+2)}) edges.
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Max-Bisections of graphs without even cycles
Every C_{2k}-free graph with minimum degree at least k has a balanced bipartition with at least m/2 + Omega(m^{(2k+1)/(2k+2)}) edges.