In the paper Hall's theorem for hypergraphs (Aharoni and Haxell, 2000), the authors prove a theorem on the existence of perfect matchings in bipartite hypergraphs, using Sperner's lemma. At the last page (6), they say that "we have here a topological proof of Hall's theorem" (for bipartite graphs). I thought it should be easy to write this proof explicitly, since a simple bipartite graph is just a bipartite hypergraph in which each hyperedge is of size 2. But there is a problem: during the proof, the authors assume that the sets of neighbors (i.e., the sets $N(x)$ for each vertex $x\in X$, where $X$ is one part of the hypergraph) are pairwise-disjoint. For a hypergraph, this assumption is without loss of generality, since we can add dummy vertices to edges, and this does not affect the theorem conditions or concludion. But in a graph, we cannot add vertices to edges.

So my question is: is there an explicitly-written proof of Hall's marriage theorem, using Sperner's lemma (or a similar topological theorem)?