Tight bounds for triangle-free induced subgraphs under k-domination

Determine a tight upper bound on the maximum order αF(G) of a triangle-free induced subgraph of a claw-free graph G in terms of the k-domination number γk(G) for every integer k ≥ 4.

Background

Theorem 11 proves that for k ∈ {1, 2, 3}, a claw-free graph G and the family F of triangle-free graphs satisfy αF(G) ≤ (5/k)γk(G), with the bound sharp. The authors state that the corresponding tight bound for k ≥ 4 remains unresolved. They mention a bound obtainable from the proof ideas, but explicitly note that its attainability is unlikely; the open problem is therefore to determine the true sharp constant.