Written on the Wall II - Conjecture 322

Reference: E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc

theorem WrittenOnTheWallII.GraphConjecture322.conjecture322 {α : Type u_1} [Fintype α] [DecidableEq α] (G : SimpleGraph α) [DecidableRel G.Adj] (hG : G.Connected) (hn : 5 ≤ Fintype.card α) (h : ∀ (v : α), G.indepNeighborsCard v ≤ 1) : G.IsWellTotallyDominated

WOWII Conjecture 322

Let G be a simple connected graph on n ≥ 5 vertices. If the maximum over all vertices v of l(v) — the independence number of the neighborhood N(v) of v — is at most 1, then G is well totally dominated.

Here l(v) = α(G[N(v)]) is the independence number of the subgraph induced by the open neighborhood of v.