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A hierarchy of randomness for graphs

Probability & statistics · math.CO · math.PR · posed by Miklós Simonovits, Vera T. Sós · open

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Statement

Is it true that fDR2(n,C)≈32nf_{DR}^{2}(n,\mathscr{C})\approx\frac{3}{2}n ? If not, can one prove at least, that fDR2(n,C)<(2−c)n?(34)f_{\mathrm{DR}}^{2}(n,\mathcal{C})<(2-c)n?\quad(34)

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  • A hierarchy of randomness for graphs
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  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: Reconstructed statement: C\mathcal C is the graph property “contains a cycle.” fDR2(n,C)f_{\mathrm{DR}}^{2}(n,\mathcal C) is the deterministic-random threshold: the least edge number forcing every nn-vertex graph GG, under a uniformly random red/blue edge-colouring, to contain a monochromatic cycle with probability 1−o(1)1-o(1). Problem 5.4 asks whether

    fDR2(n,C)∼32n,f_{\mathrm{DR}}^{2}(n,\mathcal C)\sim \frac32 n,

    and, failing that, whether there is a constant c>0c>0 such that

    fDR2(n,C)<(2−c)n.f_{\mathrm{DR}}^{2}(n,\mathcal C)<(2-c)n .

    Result: The weaker assertion is true. In fact,

    fDR2(n,C)≤1910n+O(1),f_{\mathrm{DR}}^{2}(n,\mathcal C)\le \frac{19}{10}n+O(1),

    so (34) holds, for example with c=1/20c=1/20.

    Proof. Let GG be an nn-vertex graph with m≥1.9nm\ge 1.9n edges. If m>2n−2m>2n-2, then every red/blue colouring has a monochromatic cycle, since two forests on the same nn vertices have at most 2(n−1)2(n-1) edges total.

    Assume m≤2nm\le 2n. Randomly colour each edge red or blue. Let IRI_R and IBI_B be the numbers of vertices isolated in the red and blue subgraphs, and put I=IR+IBI=I_R+I_B. If both colour classes are forests, with component counts kR,kBk_R,k_B, then

    kR+kB=(n−eR)+(n−eB)=2n−m,k_R+k_B=(n-e_R)+(n-e_B)=2n-m,

    and since isolated vertices are components,

    I≤kR+kB=2n−m.(1)I\le k_R+k_B=2n-m. \tag{1}

    For a vertex vv of degree d(v)d(v),

    Pr⁡(v is red-isolated)=Pr⁡(v is blue-isolated)=2−d(v).\Pr(v\text{ is red-isolated})=\Pr(v\text{ is blue-isolated})=2^{-d(v)}.

    Hence, by convexity of x↦2−xx\mapsto 2^{-x},

    EI=2∑v2−d(v)≥2n 2−2m/n.\mathbb E I=2\sum_v 2^{-d(v)} \ge 2n\,2^{-2m/n}.

    Writing x=m/n∈[1.9,2]x=m/n\in[1.9,2],

    21−2x−(2−x)≥2−14/5−0.1=:η>0.2^{1-2x}-(2-x)\ge 2^{-14/5}-0.1=: \eta>0.

    Thus

    EI≥(2n−m)+ηn.\mathbb E I\ge (2n-m)+\eta n.

    Changing one edge colour changes II by at most 22. McDiarmid’s inequality gives

    Pr⁡(I≤2n−m)≤Pr⁡(I≤EI−ηn)≤exp⁡(−Ω(n)).\Pr(I\le 2n-m) \le \Pr(I\le \mathbb E I-\eta n) \le \exp(-\Omega(n)).

    By (1), with probability 1−exp⁡(−Ω(n))1-\exp(-\Omega(n)), the two colour classes cannot both be forests. Therefore some colour contains a cycle.

    Thus every nn-vertex graph with at least 1.9n1.9n edges has a monochromatic cycle in a random two-colouring with probability 1−o(1)1-o(1). This proves

    fDR2(n,C)≤1.9n+O(1)<(2−c)nf_{\mathrm{DR}}^{2}(n,\mathcal C)\le 1.9n+O(1)<(2-c)n

    for some absolute c>0c>0. The sharper 32n\frac32 n asymptotic is not settled by this argument.

    Citation: Original problem: M. Simonovits and V. T. Sós, “A hierarchy of randomness for graphs,” Discrete Mathematics, doi:10.1016/j.disc.2004.12.024, Problem 5.4. The proof above uses only elementary forest counting, Jensen’s inequality, and bounded-differences concentration.

  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 proof correctly establishes the weaker assertion in Problem 5.4. If a random 2-colouring has no monochromatic cycle, both colour classes are forests, forcing IR+IB≤2n−mI_R+I_B\le 2n-m. For m∈[1.9n,2n]m\in[1.9n,2n], Jensen gives E(IR+IB)≥(2n−m)+ηn\mathbb E(I_R+I_B)\ge (2n-m)+\eta n, and McDiarmid’s inequality gives an exponentially small probability of falling below 2n−m2n-m. The case m>2n−2m>2n-2 is deterministic.

    Thus every nn-vertex graph with at least 1.9n1.9n edges has a monochromatic cycle in almost every random 2-colouring, so fDR2(n,C)≤1.9n+O(1)<(2−c)nf_{\mathrm{DR}}^2(n,\mathcal C)\le 1.9n+O(1)<(2-c)n. This does not settle the sharper 3n/23n/2 asymptotic, but it rigorously answers the fallback question.

    Novelty assessment

    TYPE1

    Classification rationale: The accepted result appears genuinely new, but it is a very weak partial answer to the fallback part of Problem 5.4, not the main conjectured asymptotic fDR2(n,C)∼3n/2f_{\mathrm{DR}}^2(n,\mathcal C)\sim 3n/2. The proof is short and elementary: forest counting, Jensen’s inequality for isolated vertices, and bounded-differences concentration. This is useful as a remark/short communication to the authors, but it does not look substantial enough for a standalone combinatorics paper.

    Literature check: I found no evidence that the specific bound

    fDR2(n,C)≤1.9n+O(1)<(2−c)nf_{\mathrm{DR}}^2(n,\mathcal C)\le 1.9n+O(1)<(2-c)n

    or the fallback inequality fDR2(n,C)<(2−c)nf_{\mathrm{DR}}^2(n,\mathcal C)<(2-c)n is already in the literature. I checked the original article metadata via OpenAlex, CORE searches for the exact title, exact/near-exact notation such as fDRf_{DR}, fDRf_{\mathrm{DR}}, “deterministic-random”, “Problem 5.4”, and broader searches around random edge-colourings/monochromatic cycles/two forests. The broader hits concern different random Ramsey or randomly perturbed graph problems, not this fixed-graph random-colouring threshold. No stronger known resolution of Problem 5.4 was found.

    Citation: Original problem: Miklós Simonovits and Vera T. Sós, “A hierarchy of randomness for graphs,” Discrete Mathematics 303 (2005), 209–233, Problem 5.4. No prior citation for the stated partial resolution was found.

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