On domination in 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. Also, he conjectured that γ(H)⩽⌈n/3⌉ for every connected 3-regular (cubic) n-vertex graph H. In this note, we disprove this conjecture. We construct a connected cubic graph G on 60 vertices with γ(G)=21 and present a sequence {Gk}k=1∞ of connected cubic graphs withlimk→∞γ(Gk)|V(Gk)|⩾823=13+169.