Chordal Enhanced Power Graphs of Finite Simple Groups and Induced 4-Cycles: On Kourovka Notebook Problem 21.23

For a finite group G, the enhanced power graph E(G) has vertex set G, two distinct elements being adjacent when they generate a cyclic subgroup. Bubboloni, Fumagalli and Praeger asked (Kourovka Notebook, Problem 21.23) for which finite nonabelian simple groups E(G) has no induced cycle of a given length n ≥ 4, and for which E(G) is chordal. By their results, E(G) is chordal for PSL2(q) (q ≥ 4), the Suzuki groups, PSL3(4) and A7. We show that E(G) has an induced cycle of length at least 4 for M11, J1, the Tits group, PSL3(q) with q odd, PSU3(q) with q ≥ 3, and the Ree groups 2G2(q) with q ≥ 27; for the infinite families the cycles are given by direct constructions. Combined with the inclusion, proved by Bubboloni, Fumagalli and Praeger in an unrefereed preprint by a case analysis over the classification of finite simple groups, of all simple groups without induced 4-cycles in an explicit list, this gives: E(G) is chordal if and only if G is one of the groups of the first list. For n = 4 we show that E(G) has no induced 4-cycle for the Tits group and, using the published lists of maximal subgroups of Bray, Holt and Roney-Dougal, for PSL3(p) and PSU3(p) when p is a prime with p − 1, respectively p + 1, equal to 2^a or 3·2^a, while it has one for all other PSU3(q) and for 2G2(q) with q ≥ 27; under the same two assumptions this determines the set for n = 4. For n ≥ 5 the question remains open; we determine the least length of an induced cycle of length at least 4 for some small groups. Several steps are computer-assisted, and all certificates are included. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: KOU-21.23 (Kourovka Notebook Problem 21.23, Bubboloni–Fumagalli–Praeger: chordal enhanced power graphs of finite simple groups).