Adamus–Adamus conjecture on long cycles in balanced bipartite graphs

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Let GG be a balanced bipartite graph of order 2n2n, with minimum degree δ(G)≥r≥1\delta(G)\ge r\ge 1, where n≥2k+2rn\ge 2k+2r and k∈Zk\in\mathbb{Z}. Adamus–Adamus conjecture. If

e(G)>n(n−k−r)+r(k+r),e(G)>n(n-k-r)+r(k+r),

then GG contains a cycle of length 2n−2k2n-2k. This is presented as the criterion proposed by Adamus and Adamus for guaranteeing a prescribed long cycle; the supplied text does not state whether it has been proved or remains open.

References

Primary source

Changchang Dong, Mei Lu, Jixiang Meng and Bo Ning, “The generalized Tur'an number of long cycles in graphs and bipartite graphs”, arXiv:2406.17371 (2024).

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