Higher-regularity analogue of the counterexample property

Determine whether there exists a d-regular, d-edge-colourable graph \(G\), for some \(d\geq 4\), such that for every collection of \(d-1\) edges \((u_1,v_1),\ldots,(u_{d-1},v_{d-1})\in E(G)\), deleting all their endpoints leaves the independence number unchanged.

Background

The cubic examples motivate asking whether the same phenomenon occurs in higher degree. The proposed condition requires the endpoints of any d−1d-1 edges in a d-regular, d-edge-colourable graph to be simultaneously removable without decreasing the independence number.

This would provide higher-regularity analogues of the graph-theoretic structure underlying the counterexamples to Lovász's conjecture.

References

Does there exist a $d$-regular $d$-edge colourable graph $G$ where $d\geq 4$ and for all edges $(u_1,v_1),\dots, (u_{d-1}, v_{d-1}) \in E(G)$, $$ \alpha(G) = \alpha(G - {u_1,\dots,u_{d-1}, v_1,\dots, v_{d-1} })? $$

— A Counterexample to a Conjecture of Lovász  (2505.05339 - Clow et al., 8 May 2025) in Question 2, Section 4, Future Work