On Two Generalized Connectivities of Graphs

• Autorzy: Yuefang Sun , Fengwei Li , Zemin Jin
• Języki publikacji: EN

Abstrakty

EN

The concept of generalized k-connectivity κk(G), mentioned by Hager in 1985, is a natural generalization of the path-version of the classical connectivity. The pendant tree-connectivity τk(G) was also introduced by Hager in 1985, which is a specialization of generalized k-connectivity but a generalization of the classical connectivity. Another generalized connectivity of a graph G, named k-connectivity κ′k(G), introduced by Chartrand et al. in 1984, is a generalization of the cut-version of the classical connectivity. In this paper, we get the lower and upper bounds for the difference of κ′k(G) and τk(G) by showing that for a connected graph G of order n, if κ′k(G) ≠ n − k + 1 where k ≥ 3, then 1 ≤ κ′k(G) − τk(G) ≤ n − k; otherwise, 1 ≤ κ′k(G) ‘− τk(G) ≤ n − k + 1. Moreover, all of these bounds are sharp. We get a sharp upper bound for the 3-connectivity of the Cartesian product of any two connected graphs with orders at least 5. Especially, the exact values for some special cases are determined. Among our results, we also study the pendant tree-connectivity of Cayley graphs on Abelian groups of small degrees and obtain the exact values for τk(G), where G is a cubic or 4-regular Cayley graph on Abelian groups, 3 ≤ k ≤ n.

Słowa kluczowe

EN

k-connectivity; pendant tree-connectivity; Cartesian product; Cayley graph

Kategorie tematyczne

• 05C05: Trees
• 05C76: Graph operations (line graphs, products, etc.)
• 05C40: Connectivity

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2017
• Tom: 38
• Numer: 1
• Strony: 245-261

Daty

wydano

2018-02-01

otrzymano

2016-06-16

poprawiono

2016-11-02

zaakceptowano

2016-11-07

online

2017-12-30

Twórcy

autor

Yuefang Sun

• , , , P.R.

autor

Fengwei Li

• , , , P.R.

autor

Zemin Jin

• , , , P.R.

Identyfikatory

DOI

10.7151/dmgt.1987