GraphConjecture322.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

3 declarations, 0 contain sorry, 0 declared as axioms

Written on the Wall II - Conjecture 322

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

The source applies the local-independence hypothesis to the complement graph. Its conclusion also follows from the characterization of minimal total dominating sets of complete multipartite graphs in M. Subramanian and A. Selvakumar, Total Domination and Minimal Total Domination Polynomial of H-Join Graphs.

Imports (1)

FormalConjecturesUtil

theorem WrittenOnTheWallII.GraphConjecture322.conjecture322 WOWII Conjecture 322 Let G be a simple connected graph on n ≥ 5 vertices line 54 source
Source documentation
WOWII Conjecture 322
Let G be a simple connected graph on n ≥ 5 vertices. If the maximum over all vertices v of l(v) in the complement graph Gᶜ — the independence number of the neighborhood N(v) of v — is at most 1, then G is well totally dominated.
Here l(v) = α(Gᶜ[N(v)]) is the independence number of the subgraph induced by the open neighborhood of v in Gᶜ.
This proof was provided by Samuel Schlesinger.

L4DEL explanation
WOWII Conjecture 322 Let G be a simple connected graph on n ≥ 5 vertices. If the maximum over all vertices v of l(v) in the complement graph Gᶜ — the independence number of the neighborhood N(v) of v — is at most 1, then G is well totally dominated. Here l(v) = α(Gᶜ[N(v)]) is the independence number of the subgraph induced by the open neighborhood of v in Gᶜ. This proof was provided by Samuel Schlesinger.
Sources and attribution: Pinned Lean source (1646ca16afd6)

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, SimpleGraph.Connected, LE.le, Nat, OfNat.ofNat, Fintype.card, SimpleGraph.indepNeighborsCard, Compl.compl, SimpleGraph.IsWellTotallyDominated
Proof uses: Ne, Classical.byContradiction, Not, SimpleGraph.neighborSet, Subtype.mk, Membership.mem, Set, Eq, congrArg, Subtype, SimpleGraph.IsIndepSet, SimpleGraph.induce, Finset, Insert.insert, Singleton.singleton, Eq.mpr, Set.Pairwise, id, propext, SimpleGraph.isIndepSet_iff and 43 more
39 instances, coercions and auxiliary constants left out.

Tactic trace: 66 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture322 (G : SimpleGraph α) [DecidableRel G.Adj] (_hG : G.Connected)
    (hn : 5 ≤ Fintype.card α)
    (h : ∀ v : α, indepNeighborsCard Gᶜ v ≤ 1) :
    IsWellTotallyDominated G := by
…
```
example WrittenOnTheWallII.GraphConjecture322.(example at line 121) line 121 source
Source documentation
In K₄, all vertices have degree 3.
```
example : (⊤ : SimpleGraph (Fin 4)).maxDegree = 3 := by decide
…
```
example WrittenOnTheWallII.GraphConjecture322.(example at line 125) line 125 source
Source documentation
In the edgeless graph ⊥ on 5 vertices, the minimum degree is 0.
```
example : (⊥ : SimpleGraph (Fin 5)).minDegree = 0 := by decide
…
```