Orientations of girth-14 Cayley graphs for the 64-cycle avoidance construction

Determine whether orientations of the $H_{15}$-replacements of the girth-14 Cayley graphs on 812, 930, and 1332 vertices constructed from $mathrm{AGL}(1,p)$ for $p=23,29,31,37$ can avoid 64-cycles, thereby yielding the stated upper bounds for $f(6)$.

Background

The paper develops an H15H_{15} vertex-replacement construction for cubic base graphs. For the target cycle length 64, the orientation of each gadget determines which base-cycle expansions can attain length 64.

The authors construct girth-14 Cayley graphs on 506, 812, 930, and 1332 vertices. The 506-vertex graph is ruled out by the counting obstruction, whereas the other three pass that obstruction; the corresponding SAT orientation instances remained unresolved after several CPU-hours. A successful orientation would provide explicit bounds for f(6)f(6).

References

The corresponding SAT instances are unsatisfiable, as they must be. The same search over $\mathrm{AGL}(1,p)$ for $p=23,29,31,37$ produced girth-14 Cayley graphs on $506,812,930,1332$ vertices; the first fails Lemma~\ref{lem:count} narrowly ($16\,192>15\,939$), the others pass it, and their orientation instances (which would give $f(6)\le 12\,180$ and $13\,950$) were undecided after several CPU-hours. We leave them open.

— Small graphs without power-of-two cycles: a lower bound of 24, a correction to a construction of Exoo, and explicit bounds for f(k)  (2609.04686 - Garcia, 4 Sep 2026) in Section 5, Upper bounds for $f(k)$