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The generalized 4-connectivity of exchanged hypercubes

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  • Zhao, Shu-Li
  • Hao, Rong-Xia

Abstract

Let S ⊆ V(G) and κG(S) denote the maximum number k of edge-disjoint trees T1,T2,…,Tk in G such that V(Ti)⋂V(Tj)=S for any i,j∈{1,2,…,k} and i ≠ j. For an integer r with 2 ≤ r ≤ n, the generalized r-connectivity of a graph G is defined as κr(G)=min{κG(S)|S⊆V(G) and |S|=r}. The parameter is a generalization of traditional connectivity. So far, almost all known results of κr(G) are about regular graphs and r=3. In this paper, we focus on κr(EH(s, t)) of the exchanged hypercube for r=4, where the exchanged hypercube EH(s, t) is not regular if s ≠ t. We show that κ4(EH(s,t))=min{s,t} for min{s, t} ≥ 3. As a corollary, we obtain that κ3(EH(s,t))=min{s,t} for min{s, t} ≥ 3.

Suggested Citation

  • Zhao, Shu-Li & Hao, Rong-Xia, 2019. "The generalized 4-connectivity of exchanged hypercubes," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 342-353.
    • Handle: RePEc:eee:apmaco:v:347:y:2019:i:c:p:342-353
    DOI: 10.1016/j.amc.2018.11.023
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    References listed on IDEAS

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    1. Shasha Li & Xueliang Li, 2012. "Note on the hardness of generalized connectivity," Journal of Combinatorial Optimization, Springer, vol. 24(3), pages 389-396, October.
    2. Li, Shasha & Tu, Jianhua & Yu, Chenyan, 2016. "The generalized 3-connectivity of star graphs and bubble-sort graphs," Applied Mathematics and Computation, Elsevier, vol. 274(C), pages 41-46.
    3. Li, Hengzhe & Ma, Yingbin & Yang, Weihua & Wang, Yifei, 2017. "The generalized 3-connectivity of graph products," Applied Mathematics and Computation, Elsevier, vol. 295(C), pages 77-83.
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    Cited by:

    1. Li, Xiang-Jun & Zeng, Xue-Qian & Xu, Jun-Ming, 2019. "Generalized measures of fault tolerance for bubble sort networks," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.
    2. Niu, Ruichao & Xu, Min, 2021. "The bipanconnectivity of bipartite hypercube-like networks," Applied Mathematics and Computation, Elsevier, vol. 389(C).

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