New sufficient degree conditions for an $r$-uniform hypergraph to be $k$-edge-connected

TL;DR

This paper establishes the strongest degree conditions ensuring that an r-uniform hypergraph is k-edge-connected, super edge-connected, or maximally edge-connected, advancing hypergraph connectivity theory.

Contribution

It provides the strongest known degree conditions for r-uniform hypergraphs to be k-edge-connected, super edge-connected, and maximally edge-connected.

Findings

Derived the strongest degree condition for k-edge-connected hypergraphs.

Established the minimum degree condition for maximal edge connectivity.

Proposed an additional sufficient degree condition for k-edge-connected hypergraphs.

Abstract

An $r$-uniform hypergraphic sequence (i.e., $r$-graphic sequence) $d=(d_1, d_2,\cdots,d_n)$ is said to be forcibly $k$-edge-connected if every realization of $d$ is $k$-edge-connected. In this paper, we obtain a strongest sufficient degree condition for $d$ to be $k$-edge-connected for all $k\ge 1$ and a strongest sufficient degree condition for $d$ to be super edge-connected. As a corollary, we give the minimum degree condition for $d$ to be maximally edge-connected. We also obtain another sufficient degree condition for $d$ to be $k$-edge-connected.