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 Kn\documentclass[12pt]{minimal}
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\begin{document}$$K_n$$\end{document} of order n≥4\documentclass[12pt]{minimal}
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\begin{document}$$n \ge 4$$\end{document}, there are ⌊n+12⌋\documentclass[12pt]{minimal}
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\begin{document}$$\lfloor \frac{n+1}{2} \rfloor $$\end{document} completely independent spanning trees in L(Kn)\documentclass[12pt]{minimal}
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\begin{document}$$L(K_n)$$\end{document} where the number ⌊n+12⌋\documentclass[12pt]{minimal}
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\begin{document}$$\lfloor \frac{n+1}{2} \rfloor $$\end{document} is optimal, such that ⌊n+12⌋\documentclass[12pt]{minimal}
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\begin{document}$$\lfloor \frac{n+1}{2} \rfloor $$\end{document} completely independent spanning trees still exist in the graph obtained from L(Kn)\documentclass[12pt]{minimal}
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\begin{document}$$L(K_n)$$\end{document} by deleting any vertex (respectively, any induced path of order at most n2\documentclass[12pt]{minimal}
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\begin{document}$$\frac{n}{2}$$\end{document}) for n=4\documentclass[12pt]{minimal}
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\begin{document}$$n = 4$$\end{document} or odd n≥5\documentclass[12pt]{minimal}
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\begin{document}$$n \ge 5$$\end{document} (respectively, even n≥6\documentclass[12pt]{minimal}
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\begin{document}$$n \ge 6$$\end{document}). Concerning the connectivity and the number of completely independent spanning trees, we moreover show the following, where δ(G)\documentclass[12pt]{minimal}
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\begin{document}$$\delta (G)$$\end{document} denotes the minimum degree of G.Every 2k-connected line graph L(G) has k completely independent spanning trees if G is not super edge-connected or δ(G)≥2k\documentclass[12pt]{minimal}
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\begin{document}$$\delta (G) \ge 2k$$\end{document}.Every (4k-2)\documentclass[12pt]{minimal}
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\begin{document}$$(4k-2)$$\end{document}-connected line graph L(G) has k completely independent spanning trees if G is regular.Every (k2+2k-1)\documentclass[12pt]{minimal}
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\begin{document}$$(k^2+2k-1)$$\end{document}-connected line graph L(G) with δ(G)≥k+1\documentclass[12pt]{minimal}
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\begin{document}$$\delta (G) \ge k+1$$\end{document} has k completely independent spanning trees.
