High-girth completion from a minimum-degree condition

Determine whether a sufficiently high minimum-degree condition on the complement of a high-girth triangle packing in the complete tripartite graph K_{n,n,n} guarantees a high-girth triangle decomposition of the entire graph that contains the given packing.

Background

The paper proves a conditional high-girth completion theorem using regularity and abundance of a design-treasury. It distinguishes this result from the broader, natural completion question in which only a high minimum-degree condition is imposed on the leftover graph. The unresolved issue is that the packing and the completing decomposition must have high girth collectively; a minimum-degree hypothesis alone may not control cycles formed by triangles from both parts.

References

It is not clear if a minimum degree condition on $G$ is sufficient to ensure that there exists a high girth $K_3$-decomposition of $K_{n,n,n}$, unless perhaps if $S$ is quasi-random.

— A Combinatorial Proof of Hilton's Conjecture and Beyond  (2609.28966 - Lesgourgues et al., 24 Sep 2026) in Section 1, subsection “A High Girth Completion Theorem”