Symmetric Graphs have symmetric Matchings
Jan Fricke
TL;DR
The paper demonstrates that for bipartite graphs with an automorphism group action, the existence of a perfect matching on the graph or its factor graph implies the same for the other, especially in the case of amenable groups.
Contribution
It establishes a bidirectional relationship between perfect matchings on graphs and their factor graphs under group actions, highlighting the special case of amenable groups.
Findings
Perfect matching on the factor graph implies perfect matching on the original graph.
For amenable groups, the existence of a perfect matching on the graph guarantees one on the factor graph.
The results connect symmetry properties of graphs with group actions and matchings.
Abstract
Assume that there is a free group action of automorphisms on a bipartite graph. If there is a perfect matching on the factor graph, then obviously there is a perfect matching on the graph. Surprisingly, the reversed is also true for amenable groups: if there is a perfect matching on the graph, there is also a perfect matching on the factor graph, i. e. a group invariant ("symmetric") perfect matching on the graph.
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Taxonomy
TopicsAdvanced Graph Theory Research · Finite Group Theory Research · Limits and Structures in Graph Theory