Completely Independent Spanning Trees in Line Graphs

Completely independent spanning trees in a graphGare spanning trees ofGsuchthat for any two distinct vertices ofG, the paths between them in the spanning treesare pairwise edge-disjoint and internally vertex-disjoint. In this paper, we presenta tight lower bound on the maximum number of completely independent spanningtrees inL(G), whereL(G)denotes the line graph of a graphG. Based on a newcharacterization of a graph withkcompletely independent spanning trees, we alsoshow that for any complete graphKnof ordern=4, there are[n+1/2]completelyindependent spanning trees inL(Kn)where the number[n+1/2]is optimal, such that[n+1/2]completely independent spanning trees still exist in the graph obtained fromL(Kn)by deleting any vertex (respectively, any induced path of order at mostn2)forn=4 or oddn=5 (respectively, evenn=6). Concerning the connectivity and thenumber of completely independent spanning trees, we moreover show the following,whered(G)denotes the minimum degree of G.Every2k-connected line graphL(G)haskcompletely independent spanning treesifGis not super edge-connected ord(G)=2k.-Every(4k-2)-connected line graph L(G)h ask completely independent spanning trees ifGis regular.-Every(k2+2k-1)-connected line graph L(G)withd(G)=k+1 h ask completely independent spanning tree