Claims

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1998_01_01_shauger: Shauger (Congressus Numerantium 1998) proves the conjecture for K_{1,m}-free graphs with minimum degree at least m+1 or maximum degree at least 2m-1, and bounds the order of a claw-free counterexample; a proceedings paper.

2001_01_01_daniel_shauger: Daniel and Shauger (Congressus Numerantium 2001) prove the conjecture for planar claw-free graphs, using results of Dean and of Dean, Lesniak and Saito; a proceedings paper.

2004_01_01_markstrom: Markström (Congressus Numerantium 2004) reports an exhaustive computer search showing that every cubic graph on fewer than 30 vertices has a cycle of length 4 or 8; a proceedings paper with an author-reported computation.

2011_09_18_nowbandegani_esfandiari: A 2011 workshop result, cited in the claw-free paper of the same authors, that a bipartite counterexample has at least 32 vertices, so the conjecture holds for bipartite graphs on at most 31 vertices; a computation.

2011_09_25_nowbandegani_esfandiari_haghighi_bibak: A cubic claw-free counterexample has at least 114 vertices, a claw-free graph of minimum degree at least 3 has a cycle of length a power of two or three times one; accepted on the refereed paper in Discuss. Math. Graph Theory.

2013_04_09_heckman_krakovski: Heckman and Krakovski prove the conjecture for 3-connected cubic planar graphs by a partly computer-based discharging argument; accepted on the refereed paper in the Electronic Journal of Combinatorics.

2017_11_21_ghaffari_mostaghim: Ghaffari and Mostaghim prove the conjecture for Cayley graphs on generalized quaternion, dihedral and semidihedral groups and on groups of order p cubed; accepted on the refereed paper in Aequationes Mathematicae.

2020_10_29_liu_montgomery: Liu and Montgomery's unavoidable-sequence corollary gives every graph of average degree, hence of minimum degree, above an absolute constant a cycle of length a power of two; accepted on the refereed JAMS paper.

2021_01_01_ghasemi_varmazyar: Ghasemi and Varmazyar prove the conjecture for Cayley graphs of order twice a prime square and of order four times a prime; accepted on the refereed paper in Matematički Vesnik.

2021_09_03_gao_shan: Gao and Shan prove that every P_8-free graph of minimum degree at least three contains a cycle of length 4 or 8, confirming the conjecture for P_8-free graphs; accepted on the refereed paper in Graphs and Combinatorics.

2023_08_10_hu_shen: Hu and Shen prove that every P_10-free graph of minimum degree at least three contains a cycle of length 4 or 8, confirming the conjecture for P_10-free graphs; accepted on the refereed paper in Discrete Mathematics.

2024_10_30_hegde_sandeep_shashank: Hegde, Sandeep and Shashank prove, with a computer search, that every P_13-free graph of minimum degree at least three has a cycle of length a power of two, extending Hu and Shen's P_10-free case; an arXiv preprint.

2025_08_25_carr: Carr asserts that every graph of diameter 2 and minimum degree at least 3 contains a cycle of length 4 or 8, confirming the conjecture for diameter-2 graphs; an arXiv note reported accepted by the Bulletin of the ICA.

2026_08_02_tranquilli: Tranquilli reports a certified exhaustive computation showing that every cubic bipartite graph on at most 58 vertices has a cycle of length 4, 8 or 16, so a cubic bipartite counterexample has at least 60 vertices; arXiv.

2026_08_19_duran_ballester: A case analysis of a minimal counterexample, deposited on Zenodo as a proof of the Erdős–Gyárfás conjecture and retitled by the author, in his manuscript of 1 October 2026, a reduction leaving six outcomes open.

2026_09_04_garcia: Garcia reports a SAT search with DRAT certificates showing that every graph of minimum degree at least three on at most 23 vertices has a cycle of length 4 or 8, so a counterexample has at least 24 vertices; arXiv only.

2026_10_02_temeller: A proof posted as a GitHub issue asserts that every finite graph with minimum degree at least four and diameter at most three has a cycle of length four or eight, extending Carr's diameter-two theorem; unverified.

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