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The Crossing Lemma for Multigraphs Toth, Geza


The crossing number $cr(G)$ of a graph $G$ is the minimum number of crossings over all possible drawings of $G$ on the plane. According to the Crossing Lemma, for any simple graph $G$ with $n$ vertices and $e\ge 4n$ edges, $cr(G)\ge {1\over 64}{e^3\over n^2}$. Clearly, this result does not hold for multigraphs (graphs with parallel edges or loops). We find natural conditions that imply the analogue of the Crossing Lemma for multigraphs.

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