Borel fractional perfect matchings in quasi-transitive amenable graphs

TL;DR

The paper studies when a locally finite Borel graph with componentwise quasi-transitive amenable structure admits a Borel fractional perfect matching. It introduces a type multigraph $M$ and builds a Borel homomorphism from the original graph to $M$, then uses automorphism averaging on amenable quasi-transitive components to obtain invariant fractional perfect matchings, which can be pulled back to yield a global Borel fractional perfect matching. The results yield corollaries for Bernoulli graphs and provide a tiling-based strengthening: in Borel symmetrically tileable bipartite graphs satisfying Hall's condition, there exists a Borel fractional perfect matching with a uniform kernel bound. Collectively, these methods advance the understanding of when fractional matchings can be realized in a Borel measurable way, with implications for descriptive combinatorics and measurable tiling problems.

Abstract

We show that if a locally finite Borel graph with quasitransitive amenable components admits a fractional perfect matching, it will admit a Borel fractional perfect matching. In particular, if a countable amenable quasitransitive graph admits a fractional perfect matching then its Bernoulli graph admits a Borel fractional perfect matching.

Figures (5)

• Figure 1: An amenable quasi-transitive graph.
• Figure 2: A half-edge weighted multigraph component with a fractional perfect matching. The smaller black numbers represent the half-edge weighting, the red numbers are the fractional perfect matching.
• Figure 3: A non-amenable quasi-transitive graph. The Bernoulli graph of this graph is be symmetrically tileable.
• Figure 4: The quasi-transitive graph from Figure \ref{['fig:amenable_quasi']} with edge and vertex automorphism group orbits colored.
• Figure 5: The image of the graph in Figure \ref{['fig:colorquasi']} under the homomorphism $(\varphi_V,\varphi_E)$.

Theorems & Definitions (26)

• Theorem 1.1
• Corollary 1.1
• Theorem 1.2
• Theorem 2.1: Salvatori, 1992
• Definition 3.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3