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) ≤ [n/3] 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 {G_k} _k=1^∞ of connected cubic graphs with lim _k→∞ γ(G_k)/|V(G_k)| ≥ 1/3 +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 {G_k}_k=1^∞ of 2-connected cubic graphs with lim_k→∞ γ(G_k)/|V(G _k)| ≥ 1/3 +1/78, and a sequence {G′_ι} _ι=1∞ of connected cubic graphs where for each G′_ι we have γ(G′_ι)/ |V(G′_ι)| > 1/3 + 1/69.