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Math-1149 · mathematics · Logic

Vertex Cover Integrality Gap

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Statement

Definitions and mathematical statement

All graphs are finite, simple and undirected. A vertex cover of $G$ is a set of vertices meeting every edge; write $\operatorname{vc}(G)$ for its minimum possible cardinality. In the graph vocabulary with adjacency and equality, $C^k$ is first-order logic using at most $k$ distinct variable names, which can be reused, and counting quantifiers $\exists^{\ge m}x\,\varphi(x)$ for integer $m\ge0$. This quantifier says that at least $m$ vertices satisfy $\varphi$. Write $G\equiv^C_k H$ when $G$ and $H$ satisfy exactly the same $C^k$ sentences, with no restriction on sentence length or quantifier depth.

The proposed assertion is that for every $0<\varepsilon<1$ there are $\delta>0$ and $n_0$ such that for every integer $n\ge n_0$ there are $n$-vertex graphs $G,H$, with $\operatorname{vc}(H)>0$, satisfying \[ G\equiv^C_{\lfloor\delta n\rfloor}H, \qquad \operatorname{vc}(G)\ge(2-\varepsilon)\operatorname{vc}(H). \] Increase $n_0$ if necessary so the variable bound is positive. The positive-cover condition makes explicit the intended nontrivial ratio: two edgeless graphs would otherwise satisfy the displayed inequality vacuously. The constants may depend on $\varepsilon$, but not on $n$. The target is the exact counting-logic assertion, including its linear number of variables.

Short English statement

Can two large graphs look identical to counting logic with linearly many variables while their minimum vertex covers differ by a factor arbitrarily close to two?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1149-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Math-1149

Vertex Cover Integrality Gap

Definitions and mathematical statement

All graphs are finite, simple and undirected. A vertex cover of $G$ is a set of vertices meeting every edge; write $\operatorname{vc}(G)$ for its minimum possible cardinality. In the graph vocabulary with adjacency and equality, $C^k$ is first-order logic using at most $k$ distinct variable names, which can be reused, and counting quantifiers $\exists^{\ge m}x\,\varphi(x)$ for integer $m\ge0$. This quantifier says that at least $m$ vertices satisfy $\varphi$. Write $G\equiv^C_k H$ when $G$ and $H$ satisfy exactly the same $C^k$ sentences, with no restriction on sentence length or quantifier depth.

The proposed assertion is that for every $0<\varepsilon<1$ there are $\delta>0$ and $n_0$ such that for every integer $n\ge n_0$ there are $n$-vertex graphs $G,H$, with $\operatorname{vc}(H)>0$, satisfying \[ G\equiv^C_{\lfloor\delta n\rfloor}H, \qquad \operatorname{vc}(G)\ge(2-\varepsilon)\operatorname{vc}(H). \] Increase $n_0$ if necessary so the variable bound is positive. The positive-cover condition makes explicit the intended nontrivial ratio: two edgeless graphs would otherwise satisfy the displayed inequality vacuously. The constants may depend on $\varepsilon$, but not on $n$. The target is the exact counting-logic assertion, including its linear number of variables.

Short English statement

Can two large graphs look identical to counting logic with linearly many variables while their minimum vertex covers differ by a factor arbitrarily close to two?

Research attempt

Scope and method.
This research attempt by GPT-6 Astra Ultra is dated 2 October 2026. It preserves the catalogue statement and distinguishes elementary proofs below from cited prior work. No novelty claim or formal proof-assistant verification is made. The final paragraph states the precise remaining scope. The finite checks described below are reproducible in scripts/check_attempt_batch03_root_2026_10_02.py.
The number of variables is decisive.
We construct pairs with vertex-cover ratio arbitrarily close to two that are indistinguishable in the full two-variable counting logic, at every quantifier depth. Then we exhibit a three-variable sentence distinguishing them. This proves a restricted version and pinpoints why it cannot meet the catalogue's requirement of $\lfloor\delta n\rfloor$ variables for fixed $\delta>0$. Atserias's problem discussion [R1] distinguishes the two-variable case from stronger variable bounds; the argument below does not turn a low-variable example into the asserted linear-variable theorem.
A regular pair with a near-two cover gap.
Fix an integer $d\ge2$ and put $N_0=2d(d+1)$. Let $G_0$ consist of $2d$ disjoint copies of $K_{d+1}$, and let $H_0$ consist of $d+1$ disjoint copies of $K_{d,d}$. Both graphs have $N_0$ vertices and are $d$-regular. A cover of $K_{d+1}$ needs all but one vertex and this suffices. A cover of $K_{d,d}$ has size at least $d$ because of its perfect matching, and either part attains that bound. Consequently \[ \operatorname{vc}(G_0)=2d^2,\quad \operatorname{vc}(H_0)=d(d+1),\quad \frac{\operatorname{vc}(G_0)}{\operatorname{vc}(H_0)} =\frac{2d}{d+1}=2-\frac2{d+1}. \] For any $n\ge N_0$, write $n=qN_0+r$ with $0\le r<N_0$. Take $q$ copies of each base graph and add $r$ isolated vertices to each. The ratio is unchanged and the denominator is positive. Choosing $d$ sufficiently large gives the desired numerical ratio for any fixed positive $\varepsilon$ and every sufficiently large order, in the two-variable setting.
Why all two-variable counting sentences agree.
Partition each graph into its isolated and regular vertices. For a pair of named vertices, retain only their two classes and whether they are equal or adjacent. Induction on formulas shows that the truth of every two-variable counting formula is determined by this information, with the same answer in both graphs. Atomic formulas have that property. Boolean operations preserve it. When one variable is counted while the other is fixed at a regular vertex, the available choices consist of one equal vertex, $d$ neighbours, $qN_0-d-1$ other regular vertices, and $r$ isolates. For a fixed isolated vertex there are no neighbours, one equal isolate, $r-1$ other isolates and $qN_0$ regular vertices. These counts agree in the two graphs. Thus every counting threshold preserves the induction, as do ordinary quantifiers. Sentences have no free vertices left and obtain identical truth values. The proof applies at arbitrary depth, not only to a fixed number of refinement rounds.
The exact obstruction to using more variables.
With three variables one can state that distinct vertices $x,y,z$ are pairwise adjacent. Our graph $G$ satisfies this triangle sentence because $d+1\ge3$; $H$ does not, since its nontrivial components are bipartite. For every fixed positive $\delta$, the allowed number $\lfloor\delta n\rfloor$ eventually reaches three. The exhibited pair then fails the logical equivalence required in the problem.

There is also a useful relaxation calculation. Summing the fractional vertex-cover edge inequalities on a $d$-regular graph with $N$ vertices gives $d\sum_v x_v\ge dN/2$. The all-half assignment attains $N/2$. Isolates contribute zero. Hence both graphs have identical fractional optimum, despite their very different integral covers. Equality of this elementary relaxation does not certify equivalence in a counting logic with more variables.

Verification and final status.
The root script constructs pairs for $2\le d\le6$, several copy counts and isolated remainders, and checks the degree classes and triangle distinction exactly. The component arguments prove the cover values and the structural induction proves full two-variable equivalence. Achieving comparable gaps while hiding them from linearly many variables remains unresolved by this attempt.

Sources

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