Faudree–Fronček–Ryjáček–Locke–Langley conjecture on 2-factors in claw-free graphs

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Let GG be a claw-free graph, meaning that GG has no induced subgraph isomorphic to K1,3K_{1,3}. Let δ(G)\delta(G) denote its minimum degree and let α(G)\alpha(G) denote its independence number. A 2-factor is a 22-regular spanning subgraph. Faudree et al.'s conjecture. If

δ(G)≥α(G)+1,\delta(G)\geq \alpha(G)+1,

then GG has a 22-factor with exactly α(G)\alpha(G) cycles. This conjecture proposes a substantially smaller minimum-degree condition than earlier sufficient conditions; the source states that the bound cannot be improved for existence of such a 2-factor, but does not state that the conjecture itself has been resolved.

References

Primary source

Masaki Kashima, “Degree sum conditions and a 2-factor with a bounded number of cycles in claw-free graphs”, arXiv:2504.08268 (2025).

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