Universal lower bound for domination numbers of graph covers

Establish that there exists a constant c>0 such that, for every k-fold cover G of a graph F, the domination number satisfies γ(G) ≥ c kγ(F).

Background

The paper studies how domination parameters change when a graph G is obtained as a k-fold cover of a graph F. The authors establish several upper and lower bounds for the domination number γ(G), including bounds depending on the maximum degree and specialized estimates for 3-, 4-, and 5-regular base graphs.

The proposed conjecture asks whether the domination number of every k-fold cover admits a uniform positive lower bound proportional to both the covering multiplicity k and the domination number of the base graph. The authors note that the constant c=3/5 is the most likely candidate, but do not prove that value or otherwise establish the conjectured universal bound.