Cayley-graph version of the Lovász deletion property

Determine whether the line hypergraph of every r-regular, r-edge-colourable Cayley graph contains \(r-1\) vertices whose deletion reduces its matching number.

Background

The paper observes that the two known counterexamples to Lovász's conjecture, the Biggs–Smith graph and F168D, are not Cayley graphs. This motivates isolating Cayley graphs as a potentially positive class.

The question asks whether group structure prevents the counterexample phenomenon and guarantees the existence of r−1r-1 vertices whose deletion lowers the matching number in the associated line hypergraph.

References

Is it true that if $\mathcal{L}$ is the line hypergraph of an $r$-regular $r$-edge colourable Cayley graph, then $\mathcal{L}$ contains $r-1$ vertices whose deletion reduces its matching number?

— A Counterexample to a Conjecture of Lovász  (2505.05339 - Clow et al., 8 May 2025) in Question 3, Section 4, Future Work