Refined ell-multi-k-11-representation framework

Determine whether refining multi-k-11-representation to an ℓ-multi-k-11-representation number, in which each edge belongs to at most ℓ subgraphs, yields significant results for k equal to 0 or 1.

Background

The authors define a possible refinement in which an edge may occur in at most ℓ of the subgraphs used in the representation; strict multi-k-11-representation is the special case ℓ=1. They leave unresolved whether this refinement produces useful results for word-representation (k=0) or 1-11-representation (k=1), particularly if not all graphs are 1-11-representable.

References

Could such a refinement lead to interesting results for k = 0 (word-representation) or k = 1 (assuming not all graphs are 1-11-representable)?

— On 1-11-representability and multi-1-11-representability of graphs  (2501.13871 - Alshammari et al., 23 Jan 2025) in Section 5, Concluding remarks, page 9