Sharp hop-domination bounds for graphs of girth at least r

Determine the least value f(n,r) such that every connected graph G of sufficiently large order n, minimum degree at least 2, and girth at least r satisfies γ_h(G) ≤ f(n,r), for each integer r ≥ 4; in particular, determine the unresolved values for r ≥ 5.

Background

The paper studies upper bounds on the hop domination number γ_h(G), defined as the minimum size of a set whose vertices at distance exactly two dominate all vertices outside the set. Its main result establishes the sharp bound γ_h(G) ≤ (2/5)n for sufficiently large connected triangle-free graphs with minimum degree at least 2, which corresponds to the case of girth at least 4.

The authors ask for the analogous sharp bound when the girth condition is strengthened. They define f(n,r) as the least function for which every sufficiently large connected graph of order n, minimum degree at least 2, and girth at least r satisfies the corresponding hop-domination inequality. Large odd cycles imply the lower bound f(n,r) ≥ ⌈n/3⌉. The case r=4 is resolved in the paper by f(n,4)=(2/5)n, whereas the values for every r≥5 remain open.