Counterexamples to the Cubic Graph Domination Conjecture

Let v(G) and γ(G) denote the number of vertices and the domination number of a graph G, respectively, and let ρ(G) = γ(G)/v(G). In 1996 B. Reed conjectured that ifG is a cubic graph, then γ(G) ≤ ⌈v(G)/3⌉. In 2005 A. Kostochka and B. Stodolsky disproved this conjecture for cubic graphs of connectivity one and maintained that the conjecture may still be true for cubic 2-connected graphs. Their minimum counterexample C has 4 bridges, v(C) = 60, and γ(C) = 21. In this paper we disprove Reed’s conjecture for cubic 2-connected graphs by providing a sequence (Rk : k ≥ 3) of cubic graphs of connectivity two with ρ(Rk) = 1 3 + 1 60 , where v(Rk+1) > v(Rk) > v(R3) = 60 for k ≥ 4, and so γ(R3) = 21 and γ(Rk) − ⌈v(Rk)/3⌉ → ∞ with k → ∞. We also provide a sequence of (Lr : r ≥ 1) of cubic graphs of connectivity one with ρ(Lr) > 1 3 + 1 60 . The minimum counterexample L = L1 in this sequence is ‘better’ than C in the sense that L has 2 bridges while C has 4 bridges, v(L) = 54 < 60 = v(C), and ρ(L) = 1 3 + 1 54 > 1 3 + 1 60 = ρ(C). We also give a construction providing for every r ∈ {0, 1, 2} infinitely many cubic cyclically 4-connected Hamiltonian graphs Gr such that v(Gr) = r mod 3, r ∈ {0, 2} ⇒ γ(Gr) = ⌈v(Gr)/3⌉, and r = 1 ⇒ γ(Gr) = ⌊v(Gr)/3⌋. At last we suggest a stronger conjecture on domination in cubic 3-connected graphs.