Determining the packing number D(33,5,2)

Determine the exact value of the packing number D(33,5,2), i.e., ascertain the maximum number of edge-disjoint copies of K5 that can be packed in K33.

Background

The packing number D(33,5,2) is the largest number of K5 blocks that can be packed in K33 so that each edge appears in at most one block. The paper notes that this problem remains open and records the current bounds 48 ≤ D(33,5,2) ≤ 51.

The authors relate their K33 covering result to this packing problem by observing that a 51-block K5 packing would leave a 4-regular graph on 9 vertices as the leave; their exclusion results rule out two natural candidate leave structures (the grid and prism patterns), thereby contributing partial progress to the open packing problem.

References

Finally, we explain the relevance of the K_{33} result to the open packing problem of determining the packing number D(33,5,2).

— A SAT-based Filtering Framework for Exact Coverings of K33 by Cliques of Order 3, 4 or 5  (2603.29548 - Kovař et al., 31 Mar 2026) in Abstract; also discussed in Section 8 (Connection with the packing number D(33,5,2))