For a graph G, let f(G) be the maximum number of edges in a cut of G. For a positive integer m and a set of graphs \(\mathscr {H}\), let \(f(m,\mathscr {H})\) be the minimum possible cardinality of f(G), as G ranges over all graphs on m edges which contain no graphs in \(\mathscr {H}\). Suppose that \(r\ge 8\) is an even integer and \(\mathscr {H}=\{C_{5},C_{6},\ldots ,C_{r-1}\}\). In this paper, we prove that \(f(m,\mathscr {H})\ge m/2+cm^{\frac{r}{r+1}}\) for some real \(c>0\), which improves a result of Alon et al. We also give a general lower bound on \(f(m,\mathscr {H})\) when \(\mathscr {H}=\{K_{s}+\overline{K_{t}}\}\). This extends a result of Zeng and Hou.

对于一个图形G ^，让˚F（ģ）采用的切割边缘的最大数量ģ。对于正整数m和一组图形\（\ mathscr {H} \），令\（f（m，\ mathscr {H}）\）为f（G）的最小可能基数，因为G的范围为m个边上的所有图，\（\ mathscr {H} \）中不包含图。假设\（r \ ge 8 \）是偶数整数，并且\（\ mathscr {H} = \ {C_ {5}，C_ {6}，\ ldots，C_ {r-1} \} \）。在本文中，我们证明\（f（m，\ mathscr {H}）\ ge m / 2 + cm ^ {\ frac {r} {r + 1}} \）对于某些实\（c> 0 \），这改善了Alon等人的结果。当\（\ mathscr {H} = \ {K_ {s} + \ overline {K_ {t}} \} \）时，我们也为\（f（m，\ mathscr {H}）\）给出下界。这扩展了曾和侯的结果。