Strengthened Dirac-type minimum degree conditions guarantee that the k-switch reconfiguration graphs on perfect matchings are connected and expanders, with matching lower-bound constructions showing exponential numbers of components below certain degree thresholds.
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The paper finds the threshold probability for random subgraphs of Dirac graphs to admit Hamilton cycle transversals and derives optimal counting and packing corollaries that generalize single-graph Hamilton cycle results.
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Dirac's theorem and the switch geometry of perfect matchings
Strengthened Dirac-type minimum degree conditions guarantee that the k-switch reconfiguration graphs on perfect matchings are connected and expanders, with matching lower-bound constructions showing exponential numbers of components below certain degree thresholds.
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Robust Hamiltonicity in families of Dirac graphs
The paper finds the threshold probability for random subgraphs of Dirac graphs to admit Hamilton cycle transversals and derives optimal counting and packing corollaries that generalize single-graph Hamilton cycle results.