首页|Internally disjoint trees in the line graph and total graph of the complete bipartite graph
Internally disjoint trees in the line graph and total graph of the complete bipartite graph
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NSTL
Elsevier
Let G be a connected graph, S subset of & nbsp;& nbsp;V(G) and |S| >=& nbsp;2 , a tree T in G is called an S-tree if S subset of & nbsp;& nbsp;V(T). Two S-trees T-1 and T-2 are called internally disjoint if E(T-1) & cap;& nbsp;E(T-2) = empty set & nbsp;and V(T-1) & cap;& nbsp;V(T-2) = S. For an integer r with 2 <= r & nbsp;<=& nbsp;n, the generalized r-connectivity kappa(r)(G) of a graph G is defined as kappa(r)(G) = min{kappa(G)(S) |S subset of & nbsp;V(G) and |S| = r} , where kappa(G)(S) denotes the maximum number k of internally disjoint S-trees in G . In this paper, we consider the generalized 4-connectivity of the line graph L(K-m,K-n) and total graph T(K-m,K-n) of the complete bipartite graph K-m,K-n with 2 <=& nbsp;m <=& nbsp;n. The results that kappa(4)(L(K-m,K-n)) = m + n - 3 for 2 <= m <= 3 and kappa(4)(L(K-m,K-n)) = m + n - 4 for m >=& nbsp;4 are obtained by determining kappa(4)(K-m X K-n). In addition, we obtain that kappa(4)(T(K-m,K-m)) = delta(T(K-m,K-m)) - 2 = 2 m - 2 for m >=& nbsp;2 . These results improve the known results about the generalized 3-connectivity of L(K-m,K-n) and T(K-m,K-m) in [Appl. Math. Comput. 347 (2019) 645-652]. (C)& nbsp;2022 Elsevier Inc. All rights reserved.
NSTL
Elsevier
