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Degree Sequence Conditions for Maximally Edge-Connected and Super Edge-Connected Hypergraphs

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Abstract

Let H be a connected hypergraph with minimum degree \(\delta\) and edge-connectivity \(\lambda\). The hypergraph H is maximally edge-connected if \(\lambda = \delta\), and it is super edge-connected or super-\(\lambda\), if every minimum edge-cut consists of edges incident with some vertex. There are several degree sequence conditions, for example, Goldsmith and White (Discrete Math 23: 31–36, 1978) and Bollobás (Discrete Math 28:321–323, 1979) etc. for maximally edge-connected graphs and super-\(\lambda\) graphs. In this paper, we generalize these and some other degree sequence conditions to uniform hypergraphs.

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Authors and Affiliations

  1. College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, China

    Shuang Zhao, Yingzhi Tian & Jixiang Meng

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  1. Shuang Zhao
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  2. Yingzhi Tian
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  3. Jixiang Meng
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Correspondence to Jixiang Meng.

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The research is supported by NSFC (Nos. 11531011, 11861066), Tianshan Youth Project (2018Q066).

Appendix

Appendix

If \(r = 2\), then \(t - 1 = \delta\) and \((t - 1)\big [\left( {\begin{array}{c}n - t\\ r - 1\end{array}}\right) + \left( {\begin{array}{c}t\\ r - 1\end{array}}\right) \big ] = \delta n > \delta n - 1 = (t - 1)\big [\left( {\begin{array}{c}t - 1\\ r - 1\end{array}}\right) + \left( {\begin{array}{c}n - t - 1\\ r - 1\end{array}}\right) \big ] + r(\delta - 1) + 1\). It remains to show that the inequality holds for the case \(r \ge 3\). Since \(n \ge 2t\), we have

$$\begin{aligned} \begin{aligned}&(t - 1)\bigg [\left( {\begin{array}{c}t - 1\\ r - 2\end{array}}\right) + \left( {\begin{array}{c}n - t - 1\\ r - 2\end{array}}\right) \bigg ]\\&\quad \ge 2(t - 1)\left( {\begin{array}{c}t - 1\\ r - 2\end{array}}\right) \\&\quad = t\left( {\begin{array}{c}t - 1\\ r - 2\end{array}}\right) + \left( {\begin{array}{c}t\\ r - 1\end{array}}\right) + (t - 3)\left( {\begin{array}{c}t - 1\\ r - 2\end{array}}\right) - \left( {\begin{array}{c}t - 1\\ r - 1\end{array}}\right) . \end{aligned} \end{aligned}$$
(A1)

We proceed to show

$$\begin{aligned} \begin{aligned} (t - 3)\left( {\begin{array}{c}t - 1\\ r - 2\end{array}}\right) - \left( {\begin{array}{c}t - 1\\ r - 1\end{array}}\right) > -r. \end{aligned} \end{aligned}$$
(A2)

If \(r = 3\), \((t - 3)(t - 1) - \frac{(t - 1)(t - 2)}{2} + 3 = \frac{1}{2}t^{2} - \frac{5}{2}t + 5 > 0\) holds. If \(r \ge 4\), by \(\left( {\begin{array}{c}t\\ r - 1\end{array}}\right) > \delta \ge 1\), we have \(t > r - 1 \ge 3\) and \(\left( {\begin{array}{c}t - 1\\ r - 1\end{array}}\right) - r = (\frac{t}{r - 1} - 1)\left( {\begin{array}{c}t - 1\\ r - 2\end{array}}\right) - r < (t - 3)\left( {\begin{array}{c}t - 1\\ r - 2\end{array}}\right)\). Thus, the inequality (A2) holds. Therefore, combining the inequalities (A1) and (A2), we have

$$\begin{aligned} \begin{aligned}&(t - 1)\bigg [\left( {\begin{array}{c}n - t\\ r - 1\end{array}}\right) + \left( {\begin{array}{c}t\\ r - 1\end{array}}\right) \bigg ]\\&\quad =(t - 1)\bigg [\left( {\begin{array}{c}t - 1\\ r - 1\end{array}}\right) + \left( {\begin{array}{c}n - t - 1\\ r - 1\end{array}}\right) + \left( {\begin{array}{c}t - 1\\ r - 2\end{array}}\right) + \left( {\begin{array}{c}n - t - 1\\ r - 2\end{array}}\right) \bigg ]\\&\quad \ge (t - 1)\bigg [\left( {\begin{array}{c}t - 1\\ r - 1\end{array}}\right) + \left( {\begin{array}{c}n - t - 1\\ r - 1\end{array}}\right) \bigg ] + t\left( {\begin{array}{c}t - 1\\ r - 2\end{array}}\right) + \left( {\begin{array}{c}t\\ r - 1\end{array}}\right) \\&\qquad + (t - 3)\left( {\begin{array}{c}t - 1\\ r - 2\end{array}}\right) - \left( {\begin{array}{c}t - 1\\ r - 1\end{array}}\right) \\&\quad > (t - 1)\bigg [\left( {\begin{array}{c}t - 1\\ r - 1\end{array}}\right) + \left( {\begin{array}{c}n - t - 1\\ r - 1\end{array}}\right) \bigg ] + t\left( {\begin{array}{c}t - 1\\ r - 2\end{array}}\right) + \left( {\begin{array}{c}t\\ r - 1\end{array}}\right) - r\\&\quad = (t - 1)\bigg [\left( {\begin{array}{c}t - 1\\ r - 1\end{array}}\right) + \left( {\begin{array}{c}n - t - 1\\ r - 1\end{array}}\right) \bigg ] + r\left( {\begin{array}{c}t\\ r - 1\end{array}}\right) - r\\&\quad \ge (t - 1)\bigg [\left( {\begin{array}{c}t - 1\\ r - 1\end{array}}\right) + \left( {\begin{array}{c}n - t - 1\\ r - 1\end{array}}\right) \bigg ] + r(\delta - 1) + 1. \end{aligned} \end{aligned}$$

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Zhao, S., Tian, Y. & Meng, J. Degree Sequence Conditions for Maximally Edge-Connected and Super Edge-Connected Hypergraphs. Graphs and Combinatorics 36, 1065–1078 (2020). https://doi.org/10.1007/s00373-020-02165-w

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