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Exponential Independence in Subcubic Graphs

Probability & statistics · math.CO · math.PR · posed by Stéphane Bessy, Johannes Pardey, Dieter Rautenbach · open

2 comments

Statement

Do subcubic graphs have exponentially independent sets of linear order?

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Source
  • Exponential Independence in Subcubic Graphs
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No person has examined this. Everything below was judged by machines. say whether it holds →

  1. exploration by a model · #1

    GPT-5.5 xhigh

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
    ai discovered
    — the result was found by a model.

    NEW

    Problem: For a finite simple graph GG and S⊆V(G)S\subseteq V(G), let

    d(G,S)(u,v)=dist⁡G−(S∖{u,v})(u,v),d_{(G,S)}(u,v)=\operatorname{dist}_{G-(S\setminus\{u,v\})}(u,v),

    with value ∞\infty if no such path exists. A set SS is exponentially independent if, for every u∈Su\in S,

    ∑v∈S∖{u}21−d(G,S)(u,v)<1.\sum_{v\in S\setminus\{u\}}2^{1-d_{(G,S)}(u,v)}<1.

    The question is whether there is a constant c>0c>0 such that every finite subcubic graph GG of order nn satisfies αe(G)≥cn\alpha_e(G)\ge cn.

    This is the natural formalization of “subcubic graphs have exponentially independent sets of linear order” in the cited paper.

    Result: The answer is no. There is a sequence of connected cubic graphs GmG_m with ∣V(Gm)∣→∞|V(G_m)|\to\infty and

    αe(Gm)∣V(Gm)∣→0.\frac{\alpha_e(G_m)}{|V(G_m)|}\to 0.

    Proof sketch with full logical core:

    Let T3T_3 be the infinite cubic tree. For S⊆V(T3)S\subseteq V(T_3), define

    f(x,S)=lim sup⁡k→∞∣S∩Bk(x)∣∣Bk(x)∣.f(x,S)=\limsup_{k\to\infty}\frac{|S\cap B_k(x)|}{|B_k(x)|}.

    Key lemma. If S⊆V(T3)S\subseteq V(T_3) satisfies

    ∑v∈S∖{u}21−d(T3,S)(u,v)≤1for every u∈S,\sum_{v\in S\setminus\{u\}}2^{1-d_{(T_3,S)}(u,v)}\le 1 \quad\text{for every }u\in S,

    then f(x,S)=0f(x,S)=0 for every x∈V(T3)x\in V(T_3).

    This is the same argument as Bessy–Pardey–Rautenbach’s infinite-tree theorem, with “<1<1” replaced by “≤1\le1”. Indeed, if f(u,S)>0f(u,S)>0 for some u∈Su\in S, root T3T_3 at uu. Let XiX_i be the first vertices of SS at distance ii from uu along the rooted branches. Writing f⃗(x)\vec f(x) for the upper density of SS in the descendant cone of xx, one obtains

    f(u,S)≤13∑i≥1∑x∈Xi21−if⃗(x).f(u,S)\le \frac13\sum_{i\ge1}\sum_{x\in X_i}2^{1-i}\vec f(x).

    Also f(x,S)≥23f⃗(x)f(x,S)\ge \frac23\vec f(x). If every x∈X=⋃iXix\in X=\bigcup_i X_i had f(x,S)<2f(u,S)f(x,S)<2f(u,S), then f⃗(x)<3f(u,S)\vec f(x)<3f(u,S), forcing

    ∑i,x21−i>1,\sum_{i,x}2^{1-i}>1,

    contrary to the assumed weight bound at uu. Hence some x∈Sx\in S has f(x,S)≥2f(u,S)f(x,S)\ge2f(u,S). Iterating gives densities exceeding 11, impossible.

    Now choose connected cubic graphs GmG_m with girth →∞\to\infty, whose existence is classical. Suppose, toward contradiction, that some ε>0\varepsilon>0 and exponentially independent sets Sm⊆V(Gm)S_m\subseteq V(G_m) satisfy ∣Sm∣≥ε∣V(Gm)∣|S_m|\ge\varepsilon |V(G_m)|.

    Take a local weak limit of the rooted marked graphs (Gm,Sm,om)(G_m,S_m,o_m), where omo_m is uniformly random. Since the girth tends to infinity, the underlying limit is T3T_3. For every fixed kk,

    E∣S∩Bk(o)∣∣Bk(o)∣=lim⁡m∣Sm∣∣V(Gm)∣≥ε,\mathbb E\frac{|S\cap B_k(o)|}{|B_k(o)|} = \lim_m \frac{|S_m|}{|V(G_m)|} \ge \varepsilon,

    because in a high-girth cubic graph every radius-kk ball has size 3⋅2k−23\cdot2^k-2. Therefore, by Fatou,

    Ef(o,S)≥ε,\mathbb E f(o,S)\ge \varepsilon,

    so with positive probability f(o,S)>0f(o,S)>0.

    On the other hand, exponential independence of each SmS_m implies that no selected vertex has truncated exponential weight >1>1. This is a local property, so it passes to the limit: almost surely, every u∈Su\in S in the limiting subset of T3T_3 has total exponential weight at most 11. The key lemma then forces f(o,S)=0f(o,S)=0 almost surely, contradiction.

    Thus no positive linear lower bound exists.

    Citation: Definitions and the infinite cubic tree argument originate in Bessy, Pardey, and Rautenbach, “Exponential independence in subcubic graphs,” Discrete Mathematics 344 (2021), 112439. Existence of connected cubic graphs of arbitrarily large girth is due to Erdős and Sachs, 1963.

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE1

    PASS

    The claimed disproof addresses the correct conjecture: a high-girth cubic sequence with αe(Gm)/∣V(Gm)∣→0\alpha_e(G_m)/|V(G_m)|\to0 disproves any universal linear lower bound for subcubic graphs.

    The argument is mathematically sound: the infinite cubic-tree density lemma is valid with the non-strict ≤1\le1 bound, the marked local weak limit of high-girth cubic graphs is T3T_3, positive finite density passes to positive expected upper density in the limit, and finite exponential independence passes to the limiting truncated weight inequalities. This contradicts the tree lemma. I found no fatal gap in the proof.

    Novelty assessment

    TYPE1

    Classification rationale: The result appears genuinely new but is a short compactness/local-weak-limit corollary of Bessy–Pardey–Rautenbach’s theorem that the infinite cubic tree has no exponentially independent set of positive density, plus existence of high-girth cubic graphs. It answers the stated open question negatively, but the proof introduces little new machinery and no quantitative finite bound, so I would not regard it as a standalone standard-journal paper without additional results.

    Literature check: I found no existing paper or note stating that high-girth cubic graphs satisfy αe(G)=o(∣V(G)∣)\alpha_e(G)=o(|V(G)|), or otherwise explicitly answering the linear-order question negatively. Searches for the exact open problem phrase, “exponentially independent” with “subcubic”, “linear order”, “high/large girth”, “infinite cubic tree”, “local weak limit”, and αe(G)\alpha_e(G) returned only the Bessy–Pardey–Rautenbach paper, mirrors/talk pages, the original Jäger–Rautenbach paper, and unrelated ordinary independence/girth papers.

    Citation: Main prior source: Stéphane Bessy, Johannes Pardey, Dieter Rautenbach, “Exponential independence in subcubic graphs,” Discrete Mathematics 344 (2021), 112439; arXiv:2010.00886.
    High-girth cubic graphs: Erdős–Sachs (1963).

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