# Is the crossing number of the line graph of $K_5$ determined?

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.

• To compute the crossing number of a specific graph such as $L(K_5)$, you can try the CRWC. Mar 31, 2022 at 16:25
• Thanks! I am trying to learn how to use this software. It doesn't look easy run. I tried to apply for it, but my email never received the verification message.
– lcz
Apr 1, 2022 at 9:21