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 2k-connected line graph L(G) has k completely independent spanning trees if G is not super edge-connected or δ(G) ≥ 2k.• Every (4k−2)-connected line graph L(G) has k completely independent spanning trees if G is regular.• Every (k 2 + 2k − 1)-connected line graph L(G) with δ(G) ≥ k + 1 has k completely independent spanning trees.

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Let T 1 , T 2 , . . . , T k be spanning trees in a graph G. If for any two vertices u, v in G, the paths from u to v in T 1 , T 2 , . . . , T k are pairwise internally disjoint, then T 1 , T 2 , . . . , T k are completely independent spanning trees in G. Completely independent spanning trees can be applied to fault-tolerant communication problems in interconnection networks. In this article, we show that there are two completely independent spanning trees in any torus network. Besides, we generalize the result for the Cartesian product. In particular, we show that there are two completely independent spanning trees in the Cartesian product of any 2-connected graphs.

An L(2, 1)-labeling of a graph G is an assignment f from the vertex set V(G) to the set of nonnegative integers such that | f (x) − f (y)| ≥ 2 if x and y are adjacent and | f (x) − f (y)| ≥ 1 if x and y are at distance 2 for all x and y in V(G). A k-L(2, 1)labeling is an assignment f : V(G) → {0, . . . , k}, and the L(2, 1)-labeling problem asks the minimum k, which we denote by λ(G), among all possible assignments. It is known that this problem is NP-hard even for graphs of treewidth 2. Tree is one of a few classes for which the problem is polynomially solvable, but still only an O(∆ 4.5 n) time algorithm for a tree T has been known so far, where ∆ is the maximum degree of T and n = |V(T )|. In this paper, we first show that an existent necessary condition for λ(T ) = ∆ + 1 is also sufficient for a tree T with ∆ = Ω( √ n), which leads a linear time algorithm for computing λ(T ) under this condition. We then show that λ(T ) can be computed in O(∆ 1.5 n) time for any tree T . Combining these, we finally obtain an O(n 1.75 ) time algorithm, which greatly improves the currently best known result.

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Mader [J Graph Theory 65 (2010), 61‐69] conjectured that for any tree T of order m, every k‐connected graph G with minimum degree at least false⌊ 3 k / 2 false⌋ + m − 1 contains a subtree T ′ ≅ T such that G − V ( T ′ ) is k‐connected. In this paper, we show that for any tree T of order m, every 2‐connected graph G with minimum degree at least max { m + n ( T ) − 3 , m + 2 } contains a subtree T ′ ≅ T such that G − V ( T ′ ) is 2‐connected, where n ( T ) denotes the number of internal vertices of T. Besides, the lower bound on the minimum degree can be improved to max { m + n ( T ) / 4 + 1 / 2 , m + 2 } and m + 2 if T is a caterpillar and a quasi‐monotone caterpillar, respectively. From our results, it follows that Mader's conjecture for 2‐connected graphs is true for any tree T with n ( T ) ≤ 5, any caterpillar T c with n ( T normalc ) = 6, or any quasi‐monotone caterpillar.