Exact girth in the Moore-tree amalgamation construction

Prove that, for every connected (k,g)-cage and every admissible Moore-tree depth r≤⌊g/4⌋, the graph produced by the Moore-tree deletion and two-copy amalgamation construction has girth exactly g rather than merely girth at least g.

Background

Theorem 2.3 constructs a connected regular graph of order 2(n−|V(T_{k,r})|) with girth at least g by deleting rooted Moore trees from two copies of a (k,g)-graph and reconnecting the deficient boundary vertices. Empirically, the resulting graphs usually retain a g-cycle, but the theorem does not establish this. A universal proof would clarify when the construction produces genuine (k,g)-graphs rather than graphs of larger girth.

References

Notably, while the reader might feel this to be obviously true for all cages, we have been unable to prove such an universal claim.

— Theoretical and Computational Approaches to Determining Sets of Orders for $(k,g)$-Graphs  (2503.06466 - Eze et al., 9 Mar 2025) in Section 2, immediately after Theorem 2.3