The induced-cycle conjecture for triangle-free graphs with nonnegative Bakry–Émery curvature

Let $G$ be a triangle-free graph satisfying $CD(0,\infty)$ with both the unweighted normalized and non-normalized Laplacians. An induced $k$-cycle conjecture. $G$ contains no induced $k$-cycle for $k\geq 5$, except for $C_n$ with $n\geq 5$ and several graphs. The conjecture seeks structural restrictions on triangle-free graphs with nonnegative Bakry–Émery curvature; the source notes that many examples consist of $4$-cycles, while a complete classification is not given.