The **line graph** of an undirected graph $G$ is another graph $L(G)$ that represents the adjacencies between edges of $G$. $L(G)$ is constructed in the following way: for each edge in $G$, make a vertex in $L(G)$; for every two edges in $G$ that have a vertex in common, make an edge between their corresponding vertices in $L(G)$.

I would like to know the crossing number of a line graph of a complete graph $K_5$. Furthermore, what is crossing-minimal drawing of $L(K_5)$?

Here's what I know now and that's about it.

**$L(K_5)$ has crossing numer 3 or more.**

We can see above result in following paper.

- Kulli V R, Akka D G, Beineke L W. On line graphs with crossing number 1[J]. Journal of Graph Theory, 1979, 3(1): 87-90.]

Ps: We see easily that $L(K_5)$ is the complement of Petersen graph.