On Domination in 2-Connected Cubic Graphs

In 1996, Reed proved that the domination number, γ(G), of every n-vertex graph G with minimum degree at least 3 is at most 3n/8 and conjectured that γ(H) ≤ dn/3e for every connected 3-regular (cubic) n-vertex graph H. In [1] this conjecture was disproved by presenting a connected cubic graph G on 60 vertices with γ(G) = 21 and a sequence {Gk} ∞ k=1 of connected cubic graphs with limk→∞ γ(Gk) |V (Gk)| ≥ 13 + 1 69 . All the counter-examples, however, had cut-edges. On the other hand, in [2] it was proved that γ(G) ≤ 4n/11 for every connected cubic n-vertex graph G with at least 10 vertices. In this note we construct a sequence of graphs {Gk} ∞ k=1 of 2-connected cubic graphs with limk→∞ γ(Gk) |V (Gk)| ≥ 13 + 1 78 , and a sequence {G ′ l} ∞ l=1 of connected cubic graphs where for each Gl we have γ(G l ) |V (G l )| > 1 3 + 1 69 .