FormalConjectures/WrittenOnTheWallII/GraphConjecture7.lean

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L4DEL
Lean4 Domain Explanation Language

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

2 declarations, 1 contain sorry, 0 declared as axioms

Written on the Wall II - Conjecture 7

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

Imports (1)
  1. theorem WrittenOnTheWallII.GraphConjecture7.conjecture7 WOWII Conjecture 7 For a simple connected graph sorry line 48 source

    Source documentation

    WOWII Conjecture 7

    For a simple connected graph $G$, $L_s(G) \ge \max_v \lambda(v) - 1 + n - 2 \alpha(G)$, where $L_s(G)$ is the maximum number of leaves over all spanning trees of $G$, $n = |V(G)|$, $\alpha(G) = G.\mathrm{indepNum}$ is the independence number, and $\lambda(v) = \mathrm{indepNeighborsCard}\, G\, v$ is the independence number of the neighbourhood of $v$.

    Proved by DeLaVina, Fajtlowicz, Waller (2002).

    L4DEL explanation

    WOWII Conjecture 7 For a simple connected graph $G$, $L_s(G) \ge \max_v \lambda(v) - 1 + n - 2 \alpha(G)$, where $L_s(G)$ is the maximum number of leaves over all spanning trees of $G$, $n = |V(G)|$, $\alpha(G) = G.\mathrm{indepNum}$ is the independence number, and $\lambda(v) = \mathrm{indepNeighborsCard}\, G\, v$ is the independence number of the neighbourhood of $v$. Proved by DeLaVina, Fajtlowicz, Waller (2002).

    Sources and attribution: Pinned Lean source (1646ca16afd6)

    depends on sorry what this means

    Statement uses: Fintype, DecidableEq, Nontrivial, SimpleGraph, DecidableRel, SimpleGraph.Adj, SimpleGraph.Connected, Nat, Finset.max', Finset.image, SimpleGraph.indepNeighborsCard, Finset.univ, LE.le, Real, HSub.hSub, HAdd.hAdd, Int, OfNat.ofNat, Fintype.card, HMul.hMul and 2 more

    Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous

    42 instances, coercions and auxiliary constants left out.

    Tactic trace: 2 steps

    Open to load the checked before-and-after goal states.

    theorem conjecture7 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :
    let maxL := (Finset.univ.image (fun v => indepNeighborsCard G v)).max' (by simp)
…
  2. example WrittenOnTheWallII.GraphConjecture7.(example at line 57) line 57 source

    Source documentation

    The independence number is nonneg.

    example (G : SimpleGraph (Fin 3)) : 0 ≤ G.indepNum := Nat.zero_le _
…