Internally disjoint trees in the line graph and total graph of the complete bipartite graph

作者：

Highlights：

• The generalized k-connectivity is a generalization of traditional connectivity.

• The generalized 4-connectivity of the line graph L(Km,n) of the complete bipartite graph Km,n is determined for 1=m=n.

• The upper bound of the generalized 4-connectivity of the total graph L(Km,n) of the complete bipartite graph Km,n is obtained for 4=m=n.

• The generalized 4-connectivity of the total graph L(Km,n) of the regular complete bipartite graph Km,n is determined for m=4.

• These results generalize the known results about the generalized 3-connectivity of the line graph L(Km,n) of Km,n and the total graph T(Km,n) of the complete bipartite graph with regular case obtained by Li, Gu and Lei in [Appl. Math. Comput., 2019, 347: 645–652].

摘要

•The generalized k-connectivity is a generalization of traditional connectivity.•The generalized 4-connectivity of the line graph L(Km,n) of the complete bipartite graph Km,n is determined for 1=m=n.•The upper bound of the generalized 4-connectivity of the total graph L(Km,n) of the complete bipartite graph Km,n is obtained for 4=m=n.•The generalized 4-connectivity of the total graph L(Km,n) of the regular complete bipartite graph Km,n is determined for m=4.•These results generalize the known results about the generalized 3-connectivity of the line graph L(Km,n) of Km,n and the total graph T(Km,n) of the complete bipartite graph with regular case obtained by Li, Gu and Lei in [Appl. Math. Comput., 2019, 347: 645–652].

论文关键词：Generalized connectivity,Line graph,Total graph,Connectivity,Reliability,Graph

论文评审过程：Received 3 October 2021, Revised 22 January 2022, Accepted 26 January 2022, Available online 8 February 2022, Version of Record 8 February 2022.

论文官网地址：https://doi.org/10.1016/j.amc.2022.126990