FormalConjectures.WrittenOnTheWallII.GraphConjecture144

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import FormalConjecturesUtil

Written on the Wall II - Conjecture 144

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

namespace WrittenOnTheWallII.GraphConjecture144open SimpleGraphvariable {α : Type*} [Fintype α] [DecidableEq α] [Nontrivial α]

WOWII Conjecture 144

For a simple connected graph $G$, $\mathrm{tree}(G) \ge \mathrm{girth}(G) - 1 + \mathrm{ecc}(\mathrm{Centers})$ where $\mathrm{tree}(G)$ is the largest induced tree size, $\mathrm{girth}(G)$ is the length of the shortest cycle ($0$ if acyclic), $\mathrm{Centers} = G.\mathrm{center}$ is the set of vertices with minimum eccentricity (the center of $G$), and $\mathrm{ecc}(\mathrm{Centers})$ is the eccentricity of the center set — the maximum distance from any non-center vertex to the nearest center vertex.

Proof sketch. Let $g = \mathrm{girth}(G)$ and $e = \mathrm{ecc}(G, \mathrm{center}(G))$. The acyclic and $e = 0$ cases are immediate. If $g \le e + 2$, the formalized Bacsó--Tuza induced-path argument, with an elementary $e = 1$ case, produces an induced tree on at least $2e + 1$ vertices. Since $g - 1 + e \le 2e + 1$, this proves the result.

Suppose instead that $e + 3 \le g$, and fix a shortest cycle $C$. Removing one vertex of $C$ leaves an induced tree on $g - 1$ vertices. Analyze the connected components outside $C$. If one has at least $e$ vertices, it supplies an attached rooted induced tree of order $e$. Otherwise, let $D$ be the sum of the components' attachment depths. If $D \ge e$, select pairwise edge-separated rooted branches of total order $e$ and attach them to $C$, deleting a suitable cycle vertex. If $D < e$, shortest-cycle contact restrictions bound corresponding exclusion arcs. Their complement consists of central cycle vertices and $D$-dominates the graph, forcing $e \le D$, a contradiction.

Finite graph search was used only during discovery to test candidate structural lemmas and reject false proof templates; no finite-search result is used in the universal proof.

@[category research solved, AMS 5,
  formal_proof using lean4 at
    "https://github.com/beowulf127/wowii144-lean/blob/046429d509b28c90ee2ec38ae27c1ad377c6a5fc/WOWII144/Main.lean"]
theorem conjecture144 (G : SimpleGraph α) (h : G.Connected) :
    (G.girth : ℝ) - 1 + (ecc G G.center : ℝ) ≤ (largestInducedTreeSize G : ℝ) := byα:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial αG:SimpleGraph αh:G.Connected⊢ ↑G.girth - 1 + ↑(G.ecc G.center) ≤ ↑G.largestInducedTreeSize
  sorryAll goals completed! 🐙

-- Sanity checks

The largestInducedTreeSize is nonneg.

@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) : 0 ≤ largestInducedTreeSize G := Nat.zero_le _

The girth of any graph is nonneg.

@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) : 0 ≤ G.girth := Nat.zero_le _

end WrittenOnTheWallII.GraphConjecture144