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import FormalConjecturesUtilSize, Order, and Connected Domination
References:
namespace SizeOrderConnectedDominationopen SimpleGraphTheorem 2.1 of S. Mukwembi, Size, order, and connected domination, Canad. Math. Bull. 57 (2014), no. 1, 141–144 claims: if $G$ is a connected triangle-free graph of order $n$ and size $m$ with connected domination number $\gamma_c$, then $$m \le \frac{(n - \gamma_c)^2}{4} + n - 1.$$
The claim is false: the 3-dimensional hypercube $Q_3$ is a counterexample, with $n = 8$, $m = 12$ and $\gamma_c = 4$, so the asserted bound reads $12 \le (8-4)^2/4 + 8 - 1 = 11$. The gap in the paper's proof (p. 143) is the unjustified assertion that there is an edge $uv$ with $\gamma_c(G) \le \gamma_c(G - {u, v})$: in $Q_3$, removing any adjacent pair of vertices leaves a graph with connected domination number $2 < 4$.
The corollaries of the paper (Corollary 2.2 and 2.3, on leaf numbers of
triangle-free graphs) remain true; Corollary 2.2 is Graffiti.pc Conjecture 1.1,
recorded as WrittenOnTheWallII.GraphConjecture2.conjecture2.
@[category research solved, AMS 5,
formal_proof using formal_conjectures at
"https://github.com/henrykmichalewski/formal-conjectures/blob/238fcea04077aee1d63c9201aa4b4b794f3a674d/FormalConjectures/Paper/SizeOrderConnectedDomination.lean#L156"]
theorem mukwembi_theorem_2_1 : answer(False) ↔
∀ (α : Type) [Fintype α] [DecidableEq α] [Nontrivial α]
(G : SimpleGraph α) [DecidableRel G.Adj],
G.Connected → G.CliqueFree 3 →
(G.edgeFinset.card : ℝ) ≤
((Fintype.card α : ℝ) - (G.connectedDominationNumber : ℝ)) ^ 2 / 4
+ (Fintype.card α : ℝ) - 1 := ⊢ False ↔
∀ (α : Type) [inst : Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [inst_3 : DecidableRel G.Adj],
G.Connected →
G.CliqueFree 3 →
↑G.edgeFinset.card ≤ (↑(Fintype.card α) - ↑G.connectedDominationNumber) ^ 2 / 4 + ↑(Fintype.card α) - 1
All goals completed! 🐙end SizeOrderConnectedDomination