A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differ by at most 1, and its size is the number of edges which go across the two parts. In this paper, motivated by several known results of Alon, Bollobás, Krivelevich and Sudakov about Max-Cut, we study maximum bisections of graphs without short even cycles. Let G be a graph on m edges without cycles of length 4 and 6. We first extend a well-known result of Shearer on maximum cuts to bisections and show that if G has a perfect matching and degree sequence $d_1,\dots ,d_n$, then G admits a bisection of size at least $m/2+\mathrm{\Omega }\left({\sum }_{i=1}^n\sqrt{d_i}\right)$. This is tight for certain polarity graphs. Together with a technique of Nikiforov, we prove that if G also contains no cycle of length $2k\ge 6$ then G either has a large bisection or is nearly bipartite. As a corollary, if G has a matching of size $\mathrm{\Theta }(n)$, then G admits a bisection of size at least $m/2+\mathrm{\Omega }\left(m^{(2k+1)/(2k+2)}\right)$ and that this is tight for $2k\in \{6,10\}$; if G has a matching of size $o(n)$, then the bound remains valid for G with minimum degree at least 2.

图的二等分是其顶点集的二等分，其中两个部分中的顶点数量最多相差1，并且其大小是跨越两个部分的边数。在本文中，受Alon，Bollobás，Krivelevich和Sudakov关于Max-Cut的几个已知结果的启发，我们研究了没有偶数周期短的图的最大二等分。令G为m个边上的图，没有长度为4和6的循环。我们首先将众所周知的Shearer结果在最大割上扩展为两等分，并证明G是否具有理想的匹配度和阶数序列$d_{\mathrm{1个}}，\dots ，d_{ñ}$，则G接受大小至少为等分的二等分$米/2+\mathrm{\Omega }（{\sum }_{\mathrm{一世}=\mathrm{1个}}^{ñ}\sqrt{d_{\mathrm{一世}}}）$。对于某些极性图来说，这很严格。结合Nikiforov的技术，我们证明如果G也不包含长度循环$2ķ\ge 6$那么G要么有一个大二等分，要么几乎是二等分。作为推论，如果G的大小匹配$\mathrm{\Theta }（ñ）$，则G接受大小至少为等分的二等分$米/2+\mathrm{\Omega }（{米}^{（2ķ+\mathrm{1个}）/（2ķ+2）}）$ 而且这对于 $2ķ\in \{6，10\}$; 如果G的大小匹配$Ø（ñ）$，则边界对于最小度至少为2的G仍然有效。