In 1994, Erd\H{o}s and Gy\'arf\'as conjectured that every graph with minimum degree at least 3 has a cycle of length a power of 2. In 2022, Gao and Shan (Graphs and Combinatorics) proved that the conjecture is true for $P_8$-free graphs, i.e., graphs without any induced copies of a path on 8 vertices. In 2024, Hu and Shen (Discrete Mathematics) improved this result by proving that the conjecture is true for $P_{10}$-free graphs. With the aid of a computer search, we improve this further by proving that the conjecture is true for $P_{13}$-free graphs.

翻译：1994年，Erdős与Gyárfás猜想：每个最小度至少为3的图均包含一个长度为2的幂的圈。2022年，Gao与Shan（Graphs and Combinatorics）证明了该猜想对于不含$P_8$的图（即不包含任何8个顶点的路径作为诱导子图的图）成立。2024年，Hu与Shen（Discrete Mathematics）改进了这一结果，证明了该猜想对于不含$P_{10}$的图成立。借助计算机搜索，我们进一步改进了该结果，证明了该猜想对于不含$P_{13}$的图成立。