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October 20, 2025Open Access

The Hamilton cycle space of random graphs

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The condition δ(G) ≥ 3 ensures not only Hamiltonicity but the richness of the Hamilton cycle space.
In random graphs G ~ G(n,p), δ(G) ≥ 2 is sufficient for Hamiltonicity as established in classical results.
As n is odd, δ(G) ≥ 3 guarantees that Hamilton cycles span the entire cycle space of random graphs.
This finding extends existing theories in graph theory by linking cycle space properties with vertex degree conditions.

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This analysis reveals that δ(G) ≥ 3 ensures C_n(G) = C(G) in odd-n random graphs, enriching Hamilton cycle properties.

Hefetz et al. (2025) studied this question.

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