The following is a definition of the resilience of a graph w.r.t to a property $\mathcal{P}$
(Local resilience) A property $\mathcal{P}$ is said to be monotone if the property is preserved under edge addition. Given a monotone increasing property $\mathcal{P}$. The local resilience of a graph $G$ with respect to $\mathcal{P}$ is the minimum number $r$ such that by deleting at each vertex of $G$ at most $r$ edges one can obtain a graph not having $\mathcal{P}$.
In a paper i'm reading they say the following two statements 1. and 2. are equal in regards to resilience.
- What is the local resilience of the hypercube with respect to containing a perfect matching?
- In other words, what is the minimum degree of a subgraph of the hypercube containing an edge from every perfect matching?
My question is why? Surely you need the smallest maximum degree of a subgraph of the hypercube containing an edge from every perfect matching? As you will need to delete at most this many edges from each vertex to remove any perfect matching?