Completely Independent Spanning Trees in Line Graphs

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 ≥ 4 , there are ⌊ n + 1 2 ⌋ completely independent spanning trees in L ( K n ) where the number ⌊ n + 1 2 ⌋ is optimal, such that ⌊ n + 1 2 ⌋ 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 n 2 ) for n = 4 or odd n ≥ 5 (respectively, even n ≥ 6 ). Concerning the connectivity and the number of completely independent spanning trees, we moreover show the following, where δ ( G ) denotes the minimum degree of G . Every 2 k -connected line graph L ( G ) has k completely independent spanning trees if G is not super edge-connected or δ ( G ) ≥ 2 k . Every ( 4 k - 2 ) -connected line graph L ( G ) has k completely independent spanning trees if G is regular. Every ( k 2 + 2 k - 1 ) -connected line graph L ( G ) with δ ( G ) ≥ k + 1 has k completely independent spanning trees.