Is the line graph of a bipartite $2K\_2$-free graph a $2K\_2$-free graph?


Definition of line graph :click <https://fr.wikipedia.org/wiki/Line_graph>



the line graph of bipartite 2k2-free graph
No, it is not the case.
Complete bipartite graphs $K\_{n,m}$ are $2K\_2$-free. But their line graphs $L(K\_{n,m})$ are not, for $m$ and $n$ big enough. Specifically, the vertices of $L(K\_{n,n})$ may be indexed by pairs $(i,j)$, with $1\leq i,j\leq n$; two vertices $(i,j)$ and $(p,q)$ are adjacent if they have a common coordinate. E.g. with $n=4$ one has $((1,1),(1,2))$ and $((3,3),(3,4))$, two edges of $L(K\_{4,4})$ that form a $2K\_2$.


