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On Equitable Colorings of Sparse Graphs

Combinatorics · math.CO · posed by Xin Zhang · open

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Statement

If G∈GkG \in G_{k} with k ≥3 is a graph with maximum degree Δ≥(2k−1)2k−1\Delta \geq\frac{(2k-1)^{2}}{k-1} ,then G is equitably m-colorable for every m≥Δm≥\Delta .

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  • On Equitable Colorings of Sparse 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: for k≥3k\ge3, let Gk\mathcal G_k denote the finite simple graphs with maximum average degree mad⁡(G)<2k\operatorname{mad}(G)<2k. If G∈GkG\in\mathcal G_k and

    Δ(G)≥(2k−1)2k−1,\Delta(G)\ge \frac{(2k-1)^2}{k-1},

    then GG has an equitable mm-coloring for every m≥Δ(G)m\ge \Delta(G). Since

    (2k−1)2k−1=4k+1k−1,\frac{(2k-1)^2}{k-1}=4k+\frac1{k-1},

    the integer hypothesis is Δ≥4k+1\Delta\ge4k+1.

    Result: The conjecture is true.

    It is enough to prove equitable Δ\Delta-colorability, since for m≥Δ+1m\ge\Delta+1 the Hajnal–Szemerédi theorem gives an equitable mm-coloring.

    Assume, to the contrary, that GG is a smallest counterexample with Δ=Δ(G)≥4k+1\Delta=\Delta(G)\ge4k+1 and mad⁡(G)<2k\operatorname{mad}(G)<2k. Then every proper subgraph is equitably Δ\Delta-colorable: if its maximum degree is at most Δ−1\Delta-1, use Hajnal–Szemerédi; otherwise use minimality.

    We use the standard equitable-coloring adjacency lemma: if a graph FF is not equitably Δ\Delta-colorable but every proper subgraph is, and xy∈E(F)xy\in E(F) with d(x)=s≤Δ/2d(x)=s\le\Delta/2, then yy has at least Δ−s+1\Delta-s+1 neighbors of degree at least Δ−s+2\Delta-s+2.

    Let L={v:d(v)≤2k−1}L=\{v:d(v)\le2k-1\}. Since mad⁡(G)<2k\operatorname{mad}(G)<2k, such vertices exist. For x∈Lx\in L, s=d(x)≤2k−1<Δ/2s=d(x)\le2k-1<\Delta/2. Thus every neighbor yy of xx satisfies

    d(y)≥Δ−s+2d(y)\ge\Delta-s+2

    and has at most s−1s-1 neighbors in LL, because all vertices of LL have degree <Δ−s+2<\Delta-s+2.

    Discharge from initial charge d(v)d(v). Each high vertex sends to each adjacent low vertex x∈Lx\in L, with d(x)=sd(x)=s, the amount

    2k−ss.\frac{2k-s}{s}.

    A low vertex receives total 2k−s2k-s, so its final charge is 2k2k.

    For a high vertex yy, if it has no low neighbors, its final charge is at least 2k2k. Otherwise choose a low neighbor of minimum degree ss. Then yy has at most s−1s-1 low neighbors, and each receives at most (2k−s)/s(2k-s)/s. Also d(y)≥Δ−s+2d(y)\ge\Delta-s+2. Hence

    μ∗(y)≥Δ−s+2−(s−1)(2k−s)s=Δ+1−2k+2ks.\mu^*(y)\ge \Delta-s+2-\frac{(s-1)(2k-s)}s =\Delta+1-2k+\frac{2k}{s}.

    Since Δ≥4k+1\Delta\ge4k+1,

    μ∗(y)−2k≥2+2ks>0.\mu^*(y)-2k\ge 2+\frac{2k}{s}>0.

    Thus every vertex has final charge at least 2k2k. Total charge is conserved, so the average degree of GG is at least 2k2k, contradicting mad⁡(G)<2k\operatorname{mad}(G)<2k.

    Therefore GG is equitably Δ\Delta-colorable, and hence equitably mm-colorable for every m≥Δm\ge\Delta.

    Citation: Uses Hajnal–Szemerédi’s equitable coloring theorem and the equitable-coloring adjacency lemma of Kostochka–Nakprasit–Pemmaraju, “On equitable coloring of dd-degenerate graphs,” SIAM J. Discrete Math. 19 (2005), 83–95.

  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 TYPE2

    PASS

    The proof is mathematically sound modulo the cited standard equitable-coloring adjacency lemma. It actually proves the stronger statement for all graphs with mad⁡(G)<2k\operatorname{mad}(G)<2k, which covers Zhang’s Gk\mathcal G_k class. The minimal-counterexample reduction, use of Hajnal–Szemerédi for m≥Δ+1m\ge \Delta+1, and the discharging calculation are correct; isolated vertices are implicitly excluded by minimality.

    A literature check finds related weaker/special results (Zhang’s k=4k=4 case, planar/1-planar results, and Liu–Zhang 2024 with larger bounds), but no prior result proving this conjectured 4k+14k+1 threshold in full.

    Novelty assessment

    TYPE2

    Classification rationale: This appears genuinely new and publishable: it resolves Zhang’s explicit Conjecture 17 and gives a clean general maximum-average-degree threshold for equitable Δ\Delta-colorability. It is not a top-journal-scale breakthrough, since the proof is short and uses standard minimal-counterexample/discharging machinery, but it improves/clarifies known sparse-graph bounds enough for a standalone note in a graph theory/combinatorics journal.

    Literature check: I found no prior paper proving this Δ≥4k+1 \Delta\ge 4k+1 / mad⁡(G)<2k\operatorname{mad}(G)<2k statement. The closest recent result is Liu–Zhang, “Equitable coloring of sparse graphs,” arXiv:2411.19801, which proves a broader density-parameter theorem but gives only the weaker specialization Δ≥6.21d\Delta\ge 6.21d for Gd\mathcal G_d. Older Kostochka–Nakprasit/Kostochka–Nakprasit–Pemmaraju results require substantially larger degree bounds or lower average-degree assumptions. Surveys and recent planar/1-planar papers list only special cases or weaker general bounds; none contains this conjecture’s full resolution.

    Citation: Xin Zhang, “On equitable colorings of sparse graphs,” Bull. Malays. Math. Sci. Soc. 39 (2016), S257–S268, Conjecture 17. Closest related: Weichan Liu and Xin Zhang, “Equitable coloring of sparse graphs,” arXiv:2411.19801; A.V. Kostochka, K. Nakprasit, S.V. Pemmaraju, SIAM J. Discrete Math. 19 (2005), 83–95.

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