Let G be a drawing of a graph with n vertices and $$e>4n$$ e > 4 n edges, in which no two adjacent edges cross and any pair of independent edges cross at most once. According to the celebrated Crossing Lemma of Ajtai, Chvátal, Newborn, Szemerédi and Leighton, the number of crossings in G is at least $$c\,{e^3\over n^2}$$ c e 3 n 2 , for a suitable constant $$c>0$$ c > 0 . In a seminal paper, Székely generalized this result to multigraphs, establishing the lower bound $$c\,{e^3\over mn^2}$$ c e 3 m n 2 , where m denotes the maximum multiplicity of an edge in G . We get rid of the dependence on m by showing that, as in the original Crossing Lemma, the number of crossings is at least $$c'{e^3\over n^2}$$ c ′ e 3 n 2 for some $$c'>0$$ c ′ > 0 , provided that the “lens” enclosed by every pair of parallel edges in G contains at least one vertex. This settles a conjecture of Bekos, Kaufmann, and Raftopoulou.

中文翻译：

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多重图的交叉引理

设 G 是一个有 n 个顶点且 $$e>4n$$e > 4 n 条边的图，其中没有两条相邻边相交，任意一对独立边最多相交一次。根据著名的 Ajtai、Chvátal、Newborn、Szemerédi 和 Leighton 的交叉引理，G 中的交叉数至少为 $$c\,{e^3\over n^2}$$ ce 3 n 2 ，对于 a合适的常数 $$c>0$$ c > 0 。在一篇开创性的论文中，Székely 将这个结果推广到多重图，建立了下界 $$c\,{e^3\over mn^2}$$ ce 3 mn 2 ，其中 m 表示 G 中边的最大多重性。我们通过证明，如在原始 Crossing Lemma 中一样，对于某些交叉引理，交叉数至少为 $$c'{e^3\over n^2}$$ c ′ e 3 n 2，从而摆脱对 m 的依赖$$c'>0$$c ′ > 0 , 假设 G 中每对平行边所包围的“透镜”至少包含一个顶点。这解决了 Bekos、Kaufmann 和 Raftopoulou 的猜想。