A Source-Scoped Lean-Verified Proof of WOWII Conjecture 19

1. Introduction

Written on the Wall II (WOWII) is a collection of graph-theoretic conjectures generated and curated by DeLaVina’s Graffiti-style conjecturing program [1,2]. Conjecture 19 relates three classical-looking graph parameters: vertex eccentricity, local independence, and the maximum size of an induced bipartite subgraph. The official WOWII open list records the conjecture in the form

$$b\left(G\right)\ge ⌊{average}_{v\in V\left(G\right)}ecc\left(v\right)+\underset{v}{max}l\left(v\right)⌋,$$

where $l\left(v\right)$ is the independence number of the neighborhood of v [3]. The corresponding resolved list does not list Conjecture 19 as resolved [4]. Beyond its provenance, the inequality is mathematically appealing because it links a global distance parameter to a local obstruction and asks how much induced bipartite structure must already be present in a connected graph.

This paper proves the conjecture for finite nontrivial connected simple graphs, in the form recorded in the cited WOWII source materials, and verifies the argument in Lean. The claim is source-scoped: it resolves the WOWII-listed statement but does not assert a separate global priority audit of all possible literature. The strict-average branch uses the WOWII 13 diameter–local independence bound; a paper proof of that bound is included here, and the same theorem is also checked in the supplementary Lean artifact. The new mathematical step specific to WOWII 19 is the endpoint construction for the equality case $\overline{e}\left(G\right)=diam\left(G\right)$. In the notation used below, the theorem is

$$⌊\overline{e}\left(G\right)+\lambda \left(G\right)⌋\le b\left(G\right),$$

where $\overline{e}\left(G\right)$ is the average eccentricity and $\lambda \left(G\right)={max}_{v\in V\left(G\right)}\alpha \left(G\left[N\left(v\right)\right]\right)$. The proof itself is elementary in outline, but the Lean development is useful because it makes the boundary conventions precise: floors are taken after coercion to $\mathbb{R}$, neighborhood maxima are identified with finite suprema, and the induced-subgraph witness in the equality branch is checked against the formal graph definitions. The exact formal identifiers and build recipe are collected in sec:formal-verification.

2. Related Work and Provenance

The graph-theoretic terminology is standard for finite simple graphs [5,6]. Eccentricity and diameter are classical metric parameters of a connected graph, and $b\left(G\right)$ denotes the maximum order of an induced bipartite subgraph. The local parameter in WOWII 19 is

$$\lambda \left(G\right)=\underset{v\in V\left(G\right)}{max}\alpha \left(G\left[N\left(v\right)\right]\right),$$

the largest independence number among open neighborhoods. Thus the conjecture sits at the intersection of three standard types of graph parameters. The quantity $b\left(G\right)$ asks for a largest induced vertex set on which all odd cycles have been removed; equivalently, it is $\left|V\right(G\left)\right|$ minus the minimum size of an odd-cycle transversal, a vertex set whose deletion makes the graph bipartite. Odd-cycle transversals and maximum induced bipartite subgraphs are well-studied from algorithmic and polyhedral viewpoints [7,8,9]. Eccentricity and diameter are metric extremal-distance parameters, while $\lambda \left(G\right)$ is a local independence parameter at distance one; related local independence parameters at prescribed distances have also been studied in connection with global independence [10]. WOWII 19 is notable because it combines these global, metric, and local quantities in one lower bound for induced bipartite structure.

Proposition 1 

(WOWII 13 bound). For every finite nontrivial connected simple graph G,

$$diam\left(G\right)+\lambda \left(G\right)-1\le b\left(G\right).$$

Proposition 1 is precisely the theorem used in the strict-average branch of the proof, where $\overline{e}\left(G\right)<diam\left(G\right)$ yields $\lfloor \overline{e}\left(G\right)+\lambda \left(G\right)\rfloor \le diam\left(G\right)+\lambda \left(G\right)-1$. The proposition is not cited as a separately refereed publication; instead, Section 5 gives a paper proof, and the supplementary Lean file WowiiConjecture13.lean gives the corresponding formal proof of SimpleGraph.conjecture13. The WOWII 19 target module imports that file, and the recorded build elaborates both files before kernel-checking the top-level WOWII 19 theorem. The bundled WOWII13 source is also scanned for sorry, admit, axiom, and placeholder markers.

Within the source materials documented for the present manuscript, the diameter-plus-local-independence lower bound enters through the WOWII program itself [1]. The equality-case argument below is the additional observation needed to pass from the diameter bound in Proposition 1 to the average-eccentricity statement in WOWII 19: when the average eccentricity is strictly below the diameter, the floor loses at least one; when equality holds, the constructed geodesic-neighborhood witness gives a bipartite induced subgraph of the required order directly.

The literature cited above situates the constituent parameters and the induced-bipartite optimization problem. It is not a full priority audit for the exact WOWII 19 inequality or all variants involving average eccentricity, diameter, local independence, and $b\left(G\right)$. For that reason, the paper’s claim remains the source-scoped one stated in the introduction and scope section: it proves the conjecture as presented in the cited WOWII source materials and supplies a formal certificate for the theorem used here.

The formal development follows the Lean 4 and mathlib ecosystem [11,12], but the mathematical interface is straightforward. The paper’s $\lambda \left(G\right)$ is matched in Lean by the finite-supremum term sSup(Set.range(indepNeighborsG)), with the identification supplied by the lemma sSup_range_ indepNeighborsReal_eq_maxIndepNeighborsCard. In the equality branch, the key formal construction is the witness theorem

exists_diam_add_indepNeighborsCard_bipartite_witness_of_diam_geodesic_from,

which combines a diametral geodesic from a vertex v with a maximum independent set in $N\left(v\right)$. Its bipartiteness verification is carried by the auxiliary construction diam_geodesic_neighbor_path_ witness_bipartite. The top-level theorem wowii19_formal_conjectures_original_shape then rewrites these normalized ingredients into the exact FormalConjectures statement.

The provenance claim is deliberately narrow. The sources recorded for this manuscript are the official WOWII page, the open-conjecture list containing number 19, the resolved list used as a negative check, and the FormalConjectures source statement used for exact target matching. Accordingly, the manuscript claims a Lean-verified proof of the statement that these WOWII sources currently present as open [1,3,4]. It does not make the stronger bibliographic claim that no proof exists elsewhere in the literature, since that would require a separate priority audit beyond the evidence assembled here.

3. Statement

Let G be a finite nontrivial connected simple graph, where nontrivial means $\left|V\right(G\left)\right|\ge 2$. For a vertex v, write $ecc\left(v\right)$ for its eccentricity and $N\left(v\right)$ for its open neighborhood. If H is a graph, let $\alpha \left(H\right)$ denote its independence number. Thus $\alpha \left(G\right[N\left(v\right)\left]\right)$ is the maximum size of an independent set contained in the induced subgraph on $N\left(v\right)$. Define

$$\lambda \left(G\right)=\underset{v\in V\left(G\right)}{max}\alpha \left(G\left[N\left(v\right)\right]\right),\overline{e}\left(G\right)={\displaystyle \frac{1}{\left|V\right(G\left)\right|}}\sum _{v\in V\left(G\right)}ecc\left(v\right).$$

Let

$$b\left(G\right)=max\left\{\right|S|:S\subseteq V(G),G[S\left]\mathrm{is}\mathrm{bipartite}\right\}.$$

Thus $b\left(G\right)$ is the maximum order, i.e. the maximum number of vertices, of an induced bipartite subgraph of G.

Because G is finite, both extrema above are genuine maxima: the set $\left\{\alpha \right(G\left[N\right(v\left)\right]):v\in V(G\left)\right\}$ is a finite nonempty subset of $\mathbb{N}$, and the set of orders of induced bipartite subgraphs of G is likewise finite and nonempty. Since G is connected, all distances and eccentricities appearing below are finite.

Theorem 1 

(WOWII Conjecture 19). For every finite nontrivial connected simple graph G,

$$⌊\overline{e}\left(G\right)+\lambda \left(G\right)⌋\le b\left(G\right).$$

The Lean declaration checked against the FormalConjectures statement has exactly this mathematical content; the notation bridge and artifact details are given in sec:formal-verification.

4. Proof Strategy

Because G is finite, the maxima defining

$$\lambda \left(G\right)=\underset{u\in V\left(G\right)}{max}\alpha \left(G\left[N\left(u\right)\right]\right)\mathrm{and}b\left(G\right)$$

are attained. In the formal statement, $\lambda \left(G\right)$ is written as the finite supremum $sup\left\{\alpha \right(G\left[N\right(u\left)\right]):u\in V(G\left)\right\}$, realized in Lean as $sSup\left(\mathrm{range}\right(\mathrm{indepNeighbors}G\left)\right)$; the finiteness of $V\left(G\right)$ identifies this supremum with the same maximum. Put $d=diam\left(G\right)$. Since every eccentricity is at most d, the average eccentricity satisfies $\overline{e}\left(G\right)\le d$.

Table 1. The two proof branches for Theorem 1.

Case	Needed lower bound for $\mathit{b}\left(\mathit{G}\right)$	Witness or input
$\overline{e}\left(G\right)<d$	$d+\lambda \left(G\right)-1$	WOWII 13 bound, proved in Section 5
$\overline{e}\left(G\right)=d$	$d+\lambda \left(G\right)$	Diametral geodesic plus maximum independent neighborhood

Table 1. The two proof branches for Theorem 1.

Case	Needed lower bound for $\mathit{b}\left(\mathit{G}\right)$	Witness or input
$\overline{e}\left(G\right)<d$	$d+\lambda \left(G\right)-1$	WOWII 13 bound, proved in Section 5
$\overline{e}\left(G\right)=d$	$d+\lambda \left(G\right)$	Diametral geodesic plus maximum independent neighborhood

The strict-average case is purely arithmetic once one applies the diameter-local independence theorem in Proposition 1. Since d and $\lambda \left(G\right)$ are integers, $\overline{e}\left(G\right)<d$ implies

$$⌊\overline{e}\left(G\right)+\lambda \left(G\right)⌋\le d+\lambda \left(G\right)-1.$$

The WOWII 13 ingredient, stated as Proposition 1 and proved in Section 5, says that every finite nontrivial connected simple graph H satisfies

$$diam\left(H\right)+\lambda \left(H\right)-1\le b\left(H\right).$$

Applying this theorem to G gives

$$d+\lambda \left(G\right)-1\le b\left(G\right).$$

Combining the two inequalities proves the first branch.

The equality case is the new part of the proof. If $\overline{e}\left(G\right)=d$, then every vertex has eccentricity d. Choose a vertex v whose open neighborhood contains an independent set A of size $\lambda \left(G\right)$, and choose a diametral geodesic

$$v=x_0,x_1,\dots ,x_d.$$

The witness is obtained by deleting the first neighbor $x_1$ from this geodesic and adjoining A:

$$S=A\cup \{x_i:0\le i\le d,i\ne 1\}.$$

The cardinality target is then $\left|S\right|=d+\lambda \left(G\right)$.

Only three elementary shortcut facts are needed to show that $G\left[S\right]$ is bipartite. First, a diametral geodesic has no chord $x_i{x}_j$ with $j\ge i+2$, since such a chord would shorten the $x_0$–$x_d$ walk. Second, A is disjoint from the retained geodesic vertices, because every vertex of A is adjacent to $x_0$, while each retained $x_i$ with $i\ge 2$ has distance i from $x_0$. Third, no $a\in A$ is adjacent to $x_i$ for $i\ge 3$, again by shortening the diametral walk through a. Thus the only possible extra edge from A to the retained geodesic is an edge to $x_2$.

These facts support the bipartition

$$X=\left\{x_0\right\}\cup \{x_i:2\le i\le d,i\mathrm{even}\},Y=A\cup \{x_i:3\le i\le d,i\mathrm{odd}\}.$$

The geodesic parity gives independence inside the geodesic part of each class, A is independent by construction, and the only possible A–geodesic edge goes from Y to X. The full verification of these shortcut claims is given in the proof below.

At the formal level, the strict branch is discharged by the Lean theorem corresponding to Proposition 1. The passage from $\overline{e}\left(G\right)=d$ to vertexwise equality $ecc\left(u\right)=d$ is handled by

vertexEccentricityNateqdiamofaverageeqdiam,

and the equality-case witness above is packaged as

exists_diam_add_indepNeighborsCard_bipartite_witness_of_diam_geodesic_from.

5. The WOWII 13 Bound

We include the proof of Proposition 1, since this is the only nontrivial input used in the strict-average branch of WOWII 19. The argument is a diameter-path deletion construction.

Proof of Proposition 1. 

It is enough to prove the following vertexwise strengthening: for every vertex v,

$$diam\left(G\right)+\alpha \left(G\right[N\left(v\right)\left]\right)-1\le b\left(G\right).$$

Taking v to maximize $\alpha \left(G\right[N\left(v\right)\left]\right)$ then gives Proposition 1. Put $D=diam\left(G\right)$, and let $A\subseteq N\left(v\right)$ be an independent set of size $\alpha \left(G\right[N\left(v\right)\left]\right)$.

If $D\le 2$, then $\left\{v\right\}\cup A$ induces a bipartite star of order $\left|A\right|+1$, so

$$D+\left|A\right|-1\le \left|A\right|+1\le b\left(G\right).$$

Assume from now on that $D\ge 3$. Choose a diametral geodesic

$$P:x_0,x_1,\dots ,x_D.$$

For indices on this geodesic, the distance identity

$$1.7em(x_i,x_j)=|i-j|$$

holds; otherwise a shorter route between $x_i$ and $x_j$ could be spliced into P, contradicting that P has length D.

Let

$$Q=\{i\in \{0,\dots ,D\}:1.7em(v,x_i)\le 2\}.$$

The set Q has at most five elements. Indeed, if $i\le j$ are in Q, then

$$j-i=1.7em(x_i,x_j)\le 1.7em(x_i,v)+1.7em(v,x_j)\le 4.$$

Similarly, the subset

$$R=\{i\in Q:1.7em(v,x_i)\le 1\}$$

has at most three elements, since two indices in R differ by at most two.

We next choose a small add-back set $T\subseteq Q$ with three properties:

$$\left|T\right|+3\ge \left|Q\right|,\left|T\right|\le 2,(v,1.7em{x}_i)=2\mathrm{for}\mathrm{every}i\in T,$$

and all indices in T have the same parity. If $\left|Q\right|\le 3$, take $T=⌀$. If $\left|Q\right|=4$, then $Q\setminus R$ is nonempty because $\left|R\right|\le 3$, so one may choose any single index of $Q\setminus R$. If $\left|Q\right|=5$, let m and M be the minimum and maximum elements of Q. The preceding bound forces $M-m=4$. Both endpoints have distance exactly two from v; for example, if $1.7em(v,x_m)\le 1$, then $1.7em(x_m,x_M)\le 1+2=3$, contradicting $1.7em(x_m,x_M)=4$. Thus $T=\{m,M\}$ works, and its two indices have the same parity.

Define the retained index set

$$I=\left(\right\{0,\dots ,D\}\setminus Q)\cup T$$

and let $P_I=\{x_i:i\in I\}$. Since the vertices on P are distinct,

$$|P_I|=D+1-|Q|+|T|\ge D-2.$$

Moreover, A is disjoint from $P_I$: every vertex of A is adjacent to v, while a retained path vertex either lies outside Q, hence has distance greater than two from v, or belongs to T, hence has distance exactly two from v. The vertex v itself is also outside $A\cup P_I$. Therefore

$$|\left\{v\right\}\cup A\cup P_I|=1+|A|+|P_I|\ge D+|A|-1.$$

It remains to check bipartiteness. Let c be the common parity of the indices in T, taking $c=0$ when $T=⌀$, and let $d=1-c$. Put

$$P_0=\{x_i:i\in I,i\equiv c(mod2)\},P_1=\{x_i:i\in I,i\equiv d(mod2)\}.$$

Consider the partition

$$L=A\cup P_1,R=\left\{v\right\}\cup P_0$$

of $S=\left\{v\right\}\cup A\cup P_I$. Vertices of the same parity on the geodesic are nonadjacent, because any edge between two nonconsecutive geodesic vertices would shorten P. Hence $P_0$ and $P_1$ are independent. The set A is independent by construction. No vertex of A is adjacent to a vertex of $P_1$: if $x_i\in P_1$ lies outside Q, adjacency through a vertex of $A\subseteq N\left(v\right)$ would give $1.7em(v,x_i)\le 2$, while if $x_i\in T$, then i has parity c, not d. Finally, v is not adjacent to any vertex of $P_0$, since retained indices outside Q have distance greater than two from v, and retained indices in T have distance exactly two. Thus both L and R are independent, so $G\left[S\right]$ is bipartite. The order bound above gives an induced bipartite subgraph of order at least $D+\left|A\right|-1$, proving the vertexwise strengthening and hence the proposition. □

6. Proof of the Main Theorem

Lemma 1 

(Equality-case witness). Let G be a simple graph and let

$$v=x_0,x_1,\dots ,x_d=y$$

be a geodesic of length $d\ge 1$. Let $A\subseteq N\left(v\right)$ be an independent set. Put

$$S=A\cup \{x_i:0\le i\le d,i\ne 1\}.$$

Then $\left|S\right|=\left|A\right|+d$, and the induced subgraph $G\left[S\right]$ is bipartite.

Proof. 

We first record three elementary consequences of the geodesic condition.

Claim 1: A is disjoint from the retained geodesic vertices. Suppose $a\in A$ and $a=x_i$ for some $i\ne 1$. Since $A\subseteq N\left(v\right)$, the vertices v and $x_i$ are adjacent. If $i=0$, this is impossible because G is simple. If $i\ge 2$, then

$$v,x_i,x_{i+1},\dots ,x_d$$

is a v–y walk of length $1+(d-i)<d$, contradicting that the displayed v–y walk is geodesic. Hence $A\cap \{x_i:0\le i\le d,i\ne 1\}=⌀$. Since the geodesic vertices are distinct, this gives

$$\left|S\right|=\left|A\right|+\left|\{x_i:0\le i\le d,i\ne 1\}\right|=\left|A\right|+d.$$

The point $x_1$ is intentionally not excluded from A. If $x_1\in A$, then it is counted as an A-vertex rather than as a retained geodesic vertex, because $x_1$ was deleted from the geodesic part of S.

Claim 2: the retained geodesic vertices have no nonconsecutive chord. If $x_i$ were adjacent to $x_j$ with $0\le i<j\le d$ and $j\ge i+2$, then replacing the subwalk

$$x_i,x_{i+1},\dots ,x_j$$

by the single edge $x_i{x}_j$ would produce a v–y walk of length

$$i+1+(d-j)=d-(j-i)+1<d,$$

again contradicting geodesicity. Thus the retained geodesic vertices induce the path fragment $x_2{x}_3\dots x_d$, together with the isolated vertex $x_0=v$.

Claim 3: vertices of A have no retained geodesic neighbors past $x_2$. If $a\in A$ were adjacent to $x_i$ for some $i\ge 3$, then

$$v,a,x_i,x_{i+1},\dots ,x_d$$

would be a v–y walk of length $2+(d-i)<d$, impossible. Hence no vertex of A is adjacent to any $x_i$ with $i\ge 3$. The only possible edge from A to the retained geodesic, apart from the edges to $x_0$, is therefore an edge to $x_2$. When $d=1$, this statement is vacuous and the construction reduces to $S=A\cup \left\{x_0\right\}$.

Now define

$$X=\left\{x_0\right\}\cup \{x_i:2\le i\le d,i\mathrm{even}\},Y=A\cup \{x_i:3\le i\le d,i\mathrm{odd}\}.$$

Then $S=X\cup Y$, and $X\cap Y=⌀$. Claim 2 shows that X contains no edge: $x_0$ is isolated from the other retained geodesic vertices, and the remaining vertices of X have the same parity along the path fragment. Claim 2 also shows that the odd retained geodesic vertices in Y are pairwise nonadjacent. Claim 3 shows that no vertex of A is adjacent to an odd retained geodesic vertex, and A is independent by hypothesis. Thus Y is independent.

It remains only to note why this covers every edge of the induced subgraph $G\left[S\right]$. Edges among retained geodesic vertices are either consecutive path edges, hence join opposite parities, or do not exist by Claim 2. Edges from A to the retained geodesic vertices either go to $x_0$, possibly go to $x_2$, or do not exist by Claim 3; the existing possibilities both cross from Y to X. If $x_1\in A$, it lies in Y with the rest of A, so its edges to $x_0$ and possibly to $x_2$ are also crossing edges. Edges inside A do not exist. Therefore every edge of $G\left[S\right]$ crosses the partition $X\cup Y$, and $G\left[S\right]$ is bipartite. □

Proof of Theorem 1. 

Let $d=diam\left(G\right)$ and let $\lambda =\lambda \left(G\right)$. For every vertex u, we have $ecc\left(u\right)\le d$, so $\overline{e}\left(G\right)\le d$.

Assume first that $\overline{e}\left(G\right)<d$. Since $\lambda$ is an integer,

$$⌊\overline{e}\left(G\right)+\lambda ⌋\le d+\lambda -1.$$

By Proposition 1, applied to the present graph,

$$d+\lambda -1\le b\left(G\right).$$

Therefore $\lfloor \overline{e}\left(G\right)+\lambda \rfloor \le b\left(G\right)$.

It remains to consider $\overline{e}\left(G\right)=d$. Since every eccentricity is at most d and their average is d, every vertex has eccentricity d. Choose v such that $\alpha \left(G\right[N\left(v\right)\left]\right)=\lambda$, and let $A\subseteq N\left(v\right)$ be an independent set of size $\lambda$. Choose y with $1.7em(v,y)=d$, and fix a geodesic

$$v=x_0,x_1,\dots ,x_d=y.$$

By Lemma 1, the set

$$S=A\cup \{x_i:0\le i\le d,i\ne 1\}$$

induces a bipartite subgraph of order $\left|S\right|=d+\lambda$. Hence $b\left(G\right)\ge d+\lambda$. Since $\overline{e}\left(G\right)=d$, we obtain

$$⌊\overline{e}\left(G\right)+\lambda ⌋=d+\lambda \le b\left(G\right).$$

The two cases complete the proof. □

Remark 1. 

In the formal development, the strict branch is discharged by the Lean theorem corresponding to Proposition 1, while Lemma 1 is represented by the witness-construction lemma 

exists_ diam_ add_ indepNeighborsCard_ bipartite_ witness_ of_ diam_ geodesic_ from .

The formal statement uses $sSup(\mathrm{Set}.\mathrm{range}(\mathrm{indepNeighbors}G\left)\right)$ for the local term; in the finite setting this equals $\lambda \left(G\right)$.

7. Formal Verification

The Lean certificate is used here to fix the exact formal statement and to verify the auxiliary constructions that connect the two proof branches to that statement. For a finite nontrivial connected simple graph G, the maxima defining both

$$\lambda \left(G\right)=\underset{v\in V\left(G\right)}{max}\alpha \left(G\left[N\left(v\right)\right]\right)\mathrm{and}b\left(G\right)$$

exist because $V\left(G\right)$ is finite. Here $\alpha \left(G\right[N\left(v\right)\left]\right)$ denotes the independence number of the subgraph induced by the open neighborhood $N\left(v\right)$. In the formal development this local parameter is represented by the function indepNeighborsG, and the informal maximum $\lambda \left(G\right)$ is written as the finite supremum sSup(Set.range(indepNeighborsG)); the equivalence with the maximum-cardinality formulation is proved by sSup_range_indepNeighborsReal_eq_maxIndepNeighborsCard.

The notation bridge used by the formal theorem is as follows:

$b\left(G\right)$ corresponds to b G; $\lambda \left(G\right)$ corresponds to both maxIndepNeighborsCardG and sSup(Set.range(indepNeighborsG)); $\overline{e}\left(G\right)$ corresponds to the real-valued average of (eccentricityGv).toNat; the Lean theorem corresponding to the WOWII 13 bound in Proposition 1 is SimpleGraph.conjecture13.

The main normalized theorem in the file is wowii19_normalized_distEcc_maxCard, which establishes the floor inequality using the maximum local-independence cardinality directly. Its strict-average branch invokes the Lean-formalized diameter-local independence bound SimpleGraph.conjecture13. Its equality branch passes through the witness theorem

exists_diam_add_indepNeighborsCard_bipartite_witness_of_diam_geodesic_from.

Starting from a vertex v with maximum local independence, exists_indepNeighborsCard_neighbor_ indepSet extracts an independent set $A\subseteq N\left(v\right)$ of cardinality $\lambda \left(G\right)$, and diam_geodesic_neighbor_ path_witness_bipartite proves that adjoining A to a diametral geodesic after deleting the first path neighbor yields an induced bipartite witness of order at least $diam\left(G\right)+\lambda \left(G\right)$. The terminal declaration wowii19_formal_conjectures_original_shape then rewrites this normalized result into the FormalConjectures presentation by combining wowii19_distEcc_sSup_indepNeighborsReal, eccentricity_toNat_eq_vertexEccentricityNat, and the supremum-to-maximum bridge above.

The checked declaration is recorded in the target module

AmraLibrary/OpenProblemBatches/TrueOpenNextRound20260606/05_wowii_conjecture1 9.lean

with the corresponding bundled copy at result_bundle/artifacts/05_wowii_conjecture19.lean. The verified top-level theorem is wowii19_formal_conjectures_original_shape, asserting

$$⌊{\displaystyle \frac{{\sum }_{v}ecc\left(v\right)}{\left|V\right(G\left)\right|}}+sup\{\alpha \left(G\left[N\left(v\right)\right]\right):v\in V\left(G\right)\}⌋\le b\left(G\right).$$

The certificate was checked in the following environment:

Lean 4.26.0, commit d8204c9fd894f91bbb2cdfec5912ec8196fd8562

mathlib revision 2df2f0150c275ad53cb3c90f7c98ec15a56a1a67

AMRA formal repository commit e4e339e5b380375cf1c7838251966d0fc3c06929

The commit-pinned GitHub locations of the two Lean files are:

https://github.com/chainstart/amra/blob/e4e339e5b380375cf1c7838251966d0fc3c06929/amra_library/formal/AmraLibrary/OpenProblemBatches/TrueOpenNextRound20260606/05_wowii_conjecture19.lean

https://github.com/chainstart/amra/blob/e4e339e5b380375cf1c7838251966d0fc3c06929/amra_library/formal/AmraLibrary/Combinatorics/SimpleGraph/GraphConjectures/WowiiConjecture13.lean

The recorded build command is

timeout600sprlimit--as=22000000000--envLEAN_NUM_THREADS=1OMP_NUM_THREADS=1 lakeenvleanAmraLibrary/OpenProblemBatches/TrueOpenNextRound20260606/05_wowii_ conjecture19.lean

and the build report records a successful verification with no sorry, admit, axiom, or placeholder occurrences in the target proof. The memory and thread limits in the command are part of the reproducibility envelope rather than mathematical assumptions.

The trust boundary is explicit. The recorded Lean build kernel-checks the WOWII 19 target module after elaborating its imports, including the WOWII 13 theorem SimpleGraph.conjecture13, whose paper proof is given in Section 5. This kernel check is the verification step for the full dependency chain used by the target theorem. Separately, the bundled target file and bundled WOWII 13 source file were scanned for sorry, admit, axiom, and placeholder markers. The certification claim is therefore that the submitted Lean files, relative to the listed Lean/mathlib environment and the usual Lean kernel and mathlib trusted base, verify both the paper theorem and the formal counterpart of the WOWII 13 bound.

For submission, the Lean material is supplied as a supplementary artifact under

supplement/leanartifact/.

That artifact includes:

AmraLibrary/OpenProblemBatches/TrueOpenNextRound20260606/05_wowii_conjecture1 9.lean

AmraLibrary/Combinatorics/SimpleGraph/GraphConjectures/WowiiConjecture13.lean

README.md

build_reports/BUILD_RESULT.md.

The README gives SHA256 checksums, dependency revisions, and the reviewer reproduction command. Starting from the supplement root, the reviewer copies the bundled AmraLibrary/... files into a checkout of the AMRA formal workspace at the recorded commit and runs the single Lean command displayed above. Thus the formal theorem, the Lean proof of the WOWII 13 bound, and the build recipe are all present in the submission bundle.

8. Examples and Sharpness

The bound is sharp on standard graph families. For the complete graph $K_n$ with $n\ge 2$, every vertex has eccentricity 1, every open neighborhood is a clique, and hence $\lambda \left(K_n\right)=1$. The largest induced bipartite subgraph has order 2, so

$$b\left(K_n\right)=2=⌊1+1⌋=⌊\overline{e}\left(K_n\right)+\lambda \left(K_n\right)⌋.$$

Here $\overline{e}\left(K_n\right)=diam\left(K_n\right)$, so this equality is witnessed by the equality branch of the proof.

Stars give a second equality family in which the strict-average branch of the proof can be active. Let $G=K_{1,r}$ with $r\ge 1$. The whole graph is bipartite, so $b\left(G\right)=r+1$. The center has neighborhood independence r, while each leaf has neighborhood independence 1; thus $\lambda \left(G\right)=r$. The average eccentricity is

$$\overline{e}\left(G\right)={\displaystyle \frac{1+2r}{r+1}}=2-{\displaystyle \frac{1}{r+1}},$$

and therefore

$$⌊\overline{e}\left(G\right)+\lambda \left(G\right)⌋=⌊r+2-{\displaystyle \frac{1}{r+1}}⌋=r+1=b\left(G\right).$$

For $r\ge 2$, stars have $\overline{e}\left(G\right)<diam\left(G\right)$, so this sharpness family shows that the reduction through the strict-average branch can itself be tight.

The theorem is not intended as a characterization of $b\left(G\right)$. On every path and every even cycle, the whole graph is bipartite, so $b\left(G\right)=\left|V\right(G\left)\right|$, while the lower bound usually gives a much smaller value. For an odd cycle $C_{2m+1}$ with $m\ge 2$, one has $\overline{e}\left(C_{2m+1}\right)=m$, $\lambda \left(C_{2m+1}\right)=2$, and $b\left(C_{2m+1}\right)=2m$. The theorem gives $m+2\le 2m$, with equality only when $m=2$. These examples show both that the inequality can be exact and that it remains a lower bound rather than a full formula for the induced bipartite subgraph number.

9. Scope and Limitations

The theorem is stated for finite nontrivial connected simple graphs. The nontriviality assumption is present in the FormalConjectures-compatible Lean statement and avoids degenerate behavior in graph metric conventions. Connectedness is used to keep distances and eccentricities finite; disconnected graphs would require either componentwise conventions or extended-distance conventions and are outside the statement proved here.

The manuscript does not claim a new general theory of induced bipartite subgraph number. It proves the specific WOWII 19 inequality by combining the WOWII 13 diameter-local independence bound proved in Section 5 with the equality-case witness construction in Lemma 1. As explained in the introduction and related-work section, the provenance claim is source-scoped to the cited WOWII presentation rather than a global priority assertion about all literature.

10. Conclusion

We have established WOWII Conjecture 19 for finite nontrivial connected simple graphs. The argument turns on the comparison between the average eccentricity and the diameter. When $\overline{e}\left(G\right)<diam\left(G\right)$, the floor term drops by at least one, and the theorem uses the WOWII 13 diameter-local independence bound proved above. When $\overline{e}\left(G\right)=diam\left(G\right)$, every vertex is diametral, and a maximum independent set in an open neighborhood can be combined with a diametral geodesic to produce an induced bipartite subgraph of order $diam\left(G\right)+\lambda \left(G\right)$. The examples of complete graphs and stars show that the resulting inequality is sharp, while paths and cycles show that it should be read as a lower bound rather than a formula for $b\left(G\right)$. The formal development confirms the statement in Lean, with the paper proof and the certified theorem matched through the corresponding definitions of $b\left(G\right)$, $\lambda \left(G\right)$, and the floor inequality.