AI 中文总结
该研究针对Erdos-Gyarfas猜想，证明最小度≥3的图不含4-环和8-环的最小阶为24，修正了Exoo的相关构造，给出了f(k)的显式边界并验证了其最优性。
AI 中文摘要
Erdos-Gyarfas猜想指出，每个最小度至少为3的图都包含一个长度为2的幂次的环。我们通过基于SAT的穷举搜索（由DRAT证明认证）证明，顶点数不超过23、最小度至少为3的每个图都包含一个长度为4的环或一个长度为8的环；因此任何反例的顶点数至少为24，较之前发表的16的下界有所提升，且最小的最小度为3、不含4-环和8-环的图恰好有24个顶点。我们证明了Exoo针对f(5)≤450所构造的450顶点图的引理是错误的：Tutte-Coxeter图包含外边缘和弦边缘交替的8-环，且按指定构造的图包含32-环。我们修复了该构造并验证了修正后的图，因此该边界仍然成立。我们还为顶点替换构造给出了精确的窗口演算，证明对于所有k≥4，f(k)不超过最小已知围长为2的立方图的阶数的(k-2)次方的15倍加1（特别地，f(6)不超过32640，这是f(6)的首个边界），并证明Exoo针对f(4)≤78的78顶点见证在基图顶点数不超过12的小工具设计中是最优的。所有图、脚本和证明均存档于doi: https://doi.org/10.5281/zenodo.22180583。
英文摘要
The Erdos-Gyarfas conjecture states that every graph with minimum degree at least 3 contains a cycle whose length is a power of two. We prove by a SAT-based exhaustive search, certified by DRAT proofs, that every graph with minimum degree at least 3 on at most 23 vertices contains a cycle of length 4 or a cycle of length 8; consequently any counterexample has at least 24 vertices, improving the previously published bound of 16, and the smallest graph of minimum degree 3 with no 4-cycle and no 8-cycle has exactly 24 vertices. We show that the lemma underlying Exoo's 450-vertex construction for the bound f(5) at most 450 is false: the Tutte-Coxeter graph contains 8-cycles alternating between outer and chord edges, and the graph as specified contains 32-cycles. We repair the construction and verify the corrected graph, so the bound stands. We also give an exact window calculus for vertex-replacement constructions, prove that f(k) is at most 15 times the order of the smallest known cubic graph of girth 2 to the power (k-2) plus 1 for all k at least 4 (in particular f(6) is at most 32640, the first bound for f(6)), and show that Exoo's 78-vertex witness for f(4) at most 78 is optimal among gadget designs on bases with at most 12 vertices. All graphs, scripts and certificates are archived at doi:10.5281/zenodo.22180583.
Comments9 pages, 2 figures. Data, code and DRAT certificates at https://doi.org/10.5281/zenodo.22180583
