  Completely independent spanning trees in a graph $G$ are spanning trees of
$G$ such that for any two distinct vertices of $G$, the paths between them in
the spanning trees are pairwise edge-disjoint and internally vertex-disjoint.
In this paper, we present a tight lower bound on the maximum number of
completely independent spanning trees in $L(G)$, where $L(G)$ denotes the line
graph of a graph $G$. Based on a new characterization of a graph with $k$
completely independent spanning trees, we also show that for any complete graph
$K_n$ of order $n \geq 4$, there are $\lfloor \frac{n+1}{2} \rfloor$ completely
independent spanning trees in $L(K_n)$ where the number $\lfloor \frac{n+1}{2}
\rfloor$ is optimal, such that $\lfloor \frac{n+1}{2} \rfloor$ completely
independent spanning trees still exist in the graph obtained from $L(K_n)$ by
deleting any vertex (respectively, any induced path of order at most
$\frac{n}{2}$) for $n = 4$ or odd $n \geq 5$ (respectively, even $n \geq 6$).
Concerning the connectivity and the number of completely independent spanning
trees, we moreover show the following, where $\delta(G)$ denotes the minimum
degree of $G$. $\ $ $\bullet$ Every $2k$-connected line graph $L(G)$ has $k$
completely independent spanning trees if $G$ is not super edge-connected or
$\delta(G) \geq 2k$. $\ $ $\bullet$ Every $(4k-2)$-connected line graph $L(G)$
has $k$ completely independent spanning trees if $G$ is regular. $\ $ $\bullet$
Every $(k^2+2k-1)$-connected line graph $L(G)$ with $\delta(G) \geq k+1$ has
$k$ completely independent spanning trees.
