Maximum numbers of cliques in optimal 1-planar graphs

Determine, for each i in {3,4,5}, the maximum possible number of copies of the complete graph K_i in an n-vertex optimal 1-planar graph.

Background

The paper explicitly records an unresolved extremal enumeration problem for optimal 1-planar graphs. For each clique size i equal to 3, 4, or 5, the maximum number of subgraphs isomorphic to K_i that an n-vertex optimal 1-planar graph can contain is not known. Solving this problem would characterize the extremal clique counts for these three clique sizes.

References

For each $i\in{3,4,5}$, the maximum possible number of copies of $K_i$ in an $n$-vertex optimal $1$-planar graph remains unknown.

— On the Restricted Edge-Cuts of Optimal 1-Planar Graphs  (2608.13015 - Zhang et al., 13 Aug 2026) in Section 1, Introduction