10 papers analyzed
These studies suggest that while not all 4-regular graphs have perfect matchings, cubic bridgeless graphs have exponentially many perfect matchings and various bounds and algorithms exist for finding and covering edges with perfect matchings.
Bridgeless 4-regular graphs and their perfect matchings are a significant area of study in graph theory. These graphs, which have no bridges and each vertex connected to exactly four edges, present intriguing challenges and properties, particularly concerning the existence and coverage of perfect matchings.
Existence of Perfect Matchings in 4-Regular Graphs:
Perfect Matchings in Cubic Bridgeless Graphs:
Covering Edge-Sets with Perfect Matchings:
Algorithmic Advances:
Research on bridgeless 4-regular graphs and their perfect matchings reveals that while specific structures like the dual of quadrilateral meshes guarantee perfect matchings, not all 4-regular graphs share this property. In cubic bridgeless graphs, perfect matchings are abundant, with significant theoretical and algorithmic advancements supporting their study. The complexity of covering edge-sets with perfect matchings remains a challenging area, particularly highlighted by the NP-completeness of certain problems and the existence of snarks that defy simple coverage.
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