Computer-free proof for Terlep–Williford graphs

Provide a computer-free proof that the Terlep–Williford graphs TW(q) contain no 14-cycles for infinitely many prime powers q.

Background

The absence of 14-cycles in the Terlep–Williford graphs was established using a Gröbner-basis computation and the Lefschetz principle. The authors identify the derivation of a proof avoiding computer algebra as an explicit unresolved problem.

References

Problem 3. Provide a computer-free proof of the fact that the graphs TW (q) contain no 14-cycles for infinitely many q.

— Some families of graphs, hypergraphs and digraphs defined by systems of equations  (2503.07915 - Lazebnik et al., 10 Mar 2025) in Problem 3, Section 4.7