Tight independence–domination bounds for regular claw-free graphs of degree at least five

Determine a best possible upper bound on the independence number of a claw-free, d-regular graph in terms of its domination number for every integer d ≥ 5, and determine whether the upper bound in Theorem 6 is achievable for these values of d.

Background

Theorem 6 establishes that every claw-free, d-regular graph satisfies α(G) ≤ 2((d+1)/(d+2))γ(G). The paper constructs examples attaining equality for d = 2, 3, and 4. It explicitly leaves unresolved both the optimality of this bound and its attainability for all d ≥ 5.

References

However, several questions remain open. Problem 2. For d ≥ 5, determine a best possible upper bound on the independence number of a claw-free, d-regular graph G in terms of its domination number. In particular, for d ≥ 5, determine if the upper bound in Theorem 6 is achievable.

— Independence, induced subgraphs, and domination in $K_{1,r}$-free graphs  (2501.05291 - Caro et al., 9 Jan 2025) in Section 7, Problem 2; see also Section 4.1, following Theorem 6