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On the difference of two generalized connectivities of a graph

Author

Listed:
  • Yuefang Sun

    (Shaoxing University)

  • Xueliang Li

    (Nankai University)

Abstract

The concept of k-connectivity $$\kappa '_{k}(G)$$ κ k ′ ( G ) of a graph G, introduced by Chartrand in 1984, is a generalization of the cut-version of the classical connectivity. Another generalized connectivity of a graph G, named the generalized k-connectivity $$\kappa _{k}(G)$$ κ k ( G ) , mentioned by Hager in 1985, is a natural generalization of the path-version of the classical connectivity. In this paper, we get the lower and upper bounds for the difference of these two parameters by showing that for a connected graph G of order n, if $$\kappa '_k(G)\ne n-k+1$$ κ k ′ ( G ) ≠ n - k + 1 where $$k\ge 3$$ k ≥ 3 , then $$0\le \kappa '_k(G)-\kappa _k(G)\le n-k-1$$ 0 ≤ κ k ′ ( G ) - κ k ( G ) ≤ n - k - 1 ; otherwise, $$-\lfloor \frac{k}{2}\rfloor +1\le \kappa '_k(G)-\kappa _k(G)\le n-k$$ - ⌊ k 2 ⌋ + 1 ≤ κ k ′ ( G ) - κ k ( G ) ≤ n - k . Moreover, all of these bounds are sharp. Some specific study is focused for the case $$k=3$$ k = 3 . As results, we characterize the graphs with $$\kappa '_3(G)=\kappa _3(G)=t$$ κ 3 ′ ( G ) = κ 3 ( G ) = t for $$t\in \{1, n-3, n-2\}$$ t ∈ { 1 , n - 3 , n - 2 } , and give a necessary condition for $$\kappa '_3(G)=\kappa _3(G)$$ κ 3 ′ ( G ) = κ 3 ( G ) by showing that for a connected graph G of order n and size m, if $$\kappa '_3(G)=\kappa _3(G)=t$$ κ 3 ′ ( G ) = κ 3 ( G ) = t where $$1\le t\le n-3$$ 1 ≤ t ≤ n - 3 , then $$m\le {n-2\atopwithdelims ()2}+2t$$ m ≤ n - 2 2 + 2 t . Moreover, the unique extremal graph is given for the equality to hold.

Suggested Citation

  • Yuefang Sun & Xueliang Li, 2017. "On the difference of two generalized connectivities of a graph," Journal of Combinatorial Optimization, Springer, vol. 33(1), pages 283-291, January.
  • Handle: RePEc:spr:jcomop:v:33:y:2017:i:1:d:10.1007_s10878-015-9956-9
    DOI: 10.1007/s10878-015-9956-9
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    Citations

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    Cited by:

    1. Li, Hengzhe & Wang, Jiajia, 2018. "The λ3-connectivity and κ3-connectivity of recursive circulants," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 750-757.
    2. Yuefang Sun & Chenchen Wu & Xiaoyan Zhang & Zhao Zhang, 2022. "Computation and algorithm for the minimum k-edge-connectivity of graphs," Journal of Combinatorial Optimization, Springer, vol. 44(3), pages 1741-1752, October.
    3. Yuefang Sun & Chenchen Wu & Xiaoyan Zhang & Zhao Zhang, 0. "Computation and algorithm for the minimum k-edge-connectivity of graphs," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-12.

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