Hamiltonicity and cycle-length conjecture for Wenger graphs

Prove that for every m≥2 and every prime power q≥3, the Wenger graph Wm(q) contains cycles of length 2k for every integer k with 4≤k≤q^(m+1) and k≠5.

Background

Known results establish the existence of many even cycle lengths in Wenger graphs, but not the full range stated in the conjecture. The conjecture seeks a complete cycle-length coverage result subject to the known exception at k=5.

References

Conjecture 1. ([142]) For every m ≥ 2, and every prime power q, q ≥ 3, Wm(q) contains cycles of length 2k, where 4 ≤ k ≤ qm+1 and k 6 = 5.

— Some families of graphs, hypergraphs and digraphs defined by systems of equations  (2503.07915 - Lazebnik et al., 10 Mar 2025) in Conjecture 1, Section 4.4.4