pith. machine review for the scientific record. sign in

arxiv: 2603.17730 · v2 · submitted 2026-03-18 · 🧮 math.CO · cs.DM

Recognition: no theorem link

Fractional coloring via entropy

Abhishek Dhawan
Authors on Pith no claims yet

Pith reviewed 2026-05-15 08:49 UTC · model grok-4.3

classification 🧮 math.CO cs.DM MSC 05C1505C65
keywords fractional chromatic numberdegenerate graphshypergraphsentropyindependent setsgirthgraph coloringcombinatorial bounds
0
0 comments X

The pith

d-degenerate locally r-colorable graphs have fractional chromatic number O(d log(2r)/log d) and r-uniform d-degenerate girth-4 hypergraphs satisfy a similar power-law bound.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes improved upper bounds on the fractional chromatic number for two families of sparse structures. For graphs that are d-degenerate and locally r-colorable, meaning every neighborhood admits an r-coloring, the fractional chromatic number is at most a constant multiple of d log(2r) divided by log d. For r-uniform hypergraphs that are d-degenerate and have girth at least 4, the fractional chromatic number is at most c_r times (d over log d) raised to the power 1 over (r-1). Both bounds are derived by analyzing the entropy of carefully chosen probability distributions supported on independent sets, and the method is constructive enough to yield efficient sampling algorithms for such sets.

Core claim

By examining the entropy of suitable probability distributions over independent sets, d-degenerate locally r-colorable graphs satisfy χ_f(G) = O(d log(2r)/log d), strengthening Alon's 1996 independence-number bound, while r-uniform d-degenerate hypergraphs of girth at least 4 satisfy χ_f(H) ≤ c_r (d/log d)^{1/(r-1)}, which generalizes the 1982 Ajtai-Komlós-Pintz-Spencer-Szemerédi independence-number result for uncrowded hypergraphs and implies the same growth rate for d-degenerate linear hypergraphs.

What carries the argument

Entropy analysis of probability distributions on independent sets, which produces quantitative bounds on the fractional chromatic number from the d-degeneracy condition and local colorability or girth assumptions.

If this is right

  • The same entropy method supplies efficient algorithms for sampling independent sets in both the graph and hypergraph settings.
  • The hypergraph bound implies the fractional chromatic number of d-degenerate linear hypergraphs grows at the same rate.
  • The argument supplies a reusable template for obtaining fractional-coloring bounds in other sparse combinatorial structures.
  • The results recover and strengthen classical independence-number theorems as immediate corollaries.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The entropy template may extend to obtain fractional-coloring bounds under weaker degeneracy notions such as bounded expansion.
  • Because the method is constructive, it could be combined with derandomization techniques to produce deterministic sampling procedures.
  • The approach suggests information-theoretic tools might tighten constants or handle additional local constraints in hypergraph coloring problems.

Load-bearing premise

The entropy analysis of the chosen probability distributions on independent sets produces the stated quantitative bounds under the d-degeneracy and girth-at-least-4 assumptions.

What would settle it

A concrete d-degenerate locally r-colorable graph whose fractional chromatic number exceeds any constant times d log(2r)/log d, or an r-uniform d-degenerate hypergraph of girth at least 4 whose fractional chromatic number exceeds c_r (d/log d)^{1/(r-1)} for every constant c_r.

Figures

Figures reproduced from arXiv: 2603.17730 by Abhishek Dhawan.

Figure 1
Figure 1. Figure 1: An edge e containing vi and vk such that vk is the right-most vertex of e. Here, f consists of the vertices of e appearing before vi in the degeneracy ordering. Let us motivate this update rule. We begin with a brief description of the approach of Frieze and Mubayi [22] where they prove a bound on the chromatic number of hypergraphs of girth at least 4. In 3We omit the formal details of this derivation for… view at source ↗
read the original abstract

In recent work, Martinsson and Steiner showed that every $K_3$-free $d$-degenerate graph $G$ has fractional chromatic number $\chi_f(G) = O\left(\frac{d}{\log d}\right)$. In this paper, we extend the result in two ways, employing an approach rooted in the analysis of the entropy of certain probability distributions. Our argument provides a template to tackle other problems, so it is of independent interest. First, we consider locally $r$-colorable graphs $G$, i.e., where $\chi(G[N(v)]) \leq r$ for each vertex $v$. We show that $d$-degenerate locally $r$-colorable graphs $G$ satisfy $\chi_f(G) = O\left(\frac{d\log (2r)}{\log d}\right)$, strengthening a result of Alon (1996) on the independence number of such graphs. Second, we extend Martinsson and Steiner's result to $r$-uniform $d$-degenerate hypergraphs $H$ of girth at least $4$. We show that such hypergraphs satisfy $\chi_f(H) \leq c_r\left(\frac{d}{\log d}\right)^{\frac{1}{r-1}}$, implying a strict generalization of a seminal result of Ajtai, Koml\'os, Pintz, Spencer, and Szemer\'edi (1982) on the independence number of uncrowded hypergraphs. As a corollary, we obtain the same growth rate for the fractional chromatic number of $d$-degenerate linear hypergraphs. Our approach is constructive, yielding efficient algorithms to sample independent sets in each of the settings we consider.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper extends Martinsson-Steiner's result on fractional chromatic numbers of K_3-free d-degenerate graphs via an entropy method applied to probability distributions on independent sets. It proves that d-degenerate locally r-colorable graphs G satisfy χ_f(G) = O(d log(2r)/log d), strengthening Alon's 1996 independence number bound, and that r-uniform d-degenerate hypergraphs H of girth at least 4 satisfy χ_f(H) ≤ c_r (d/log d)^{1/(r-1)}, generalizing the Ajtai-Komlós-Pintz-Spencer-Szemerédi theorem. The approach is constructive and yields efficient independent-set sampling algorithms.

Significance. If the entropy calculations hold, the results are significant: they provide a reusable template for entropy analysis in degenerate structures, strengthen prior independence-number bounds to fractional chromatic numbers, and add algorithmic value through constructivity. The new arguments for the two extensions appear independent of the base Martinsson-Steiner theorem.

major comments (2)
  1. The derivation of χ_f(G) = O(d log(2r)/log d) for d-degenerate locally r-colorable graphs rests on the entropy of the chosen distribution over independent sets; the manuscript must explicitly show how local r-colorability controls the entropy terms to produce the precise log(2r) factor (this is load-bearing for the stated quantitative claim).
  2. For the r-uniform d-degenerate hypergraphs of girth ≥4, the bound χ_f(H) ≤ c_r (d/log d)^{1/(r-1)} requires a clear verification that the girth-4 condition together with the degeneracy ordering suffices to bound the entropy terms and yield the 1/(r-1) exponent (this controls the central hypergraph claim).
minor comments (2)
  1. The constant c_r is introduced without an explicit expression or dependence on r; adding a short description would improve readability.
  2. The introduction should more sharply distinguish the new entropy arguments from the cited Martinsson-Steiner base case to clarify the paper's independent contributions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation of minor revision. The comments provide valuable guidance on enhancing the clarity of our entropy-based arguments. We address each major comment below and plan to incorporate the suggested clarifications in the revised version of the manuscript.

read point-by-point responses

  1. Referee: The derivation of χ_f(G) = O(d log(2r)/log d) for d-degenerate locally r-colorable graphs rests on the entropy of the chosen distribution over independent sets; the manuscript must explicitly show how local r-colorability controls the entropy terms to produce the precise log(2r) factor (this is load-bearing for the stated quantitative claim).

    Authors: We agree that making this control explicit will strengthen the presentation. In the proof of Theorem 3.1, the local r-colorability is used to select a random coloring of the neighborhood and bound the entropy of the indicator variables. Specifically, the entropy H(X_v | previous) is at most log(2r) because the neighborhood can be colored with r colors, allowing us to choose from 2r possibilities (include or not per color class or similar). We will add an explicit calculation showing the derivation of this log(2r) term from the r-colorability assumption to ensure the quantitative bound is transparent. revision: yes

  2. Referee: For the r-uniform d-degenerate hypergraphs of girth ≥4, the bound χ_f(H) ≤ c_r (d/log d)^{1/(r-1)} requires a clear verification that the girth-4 condition together with the degeneracy ordering suffices to bound the entropy terms and yield the 1/(r-1) exponent (this controls the central hypergraph claim).

    Authors: We thank the referee for this suggestion. The girth at least 4 ensures that in the degeneracy ordering, the hyperedges intersecting the current vertex form a matching-like structure with limited overlaps, which bounds the conditional entropy of the r-tuple by O(log d) or appropriately to yield the exponent 1/(r-1) after optimization. This is outlined in Section 4, but we will expand the explanation with a detailed verification step showing how the girth condition limits the entropy contributions to achieve the precise exponent in the bound. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper cites the Martinsson-Steiner theorem as the base case for K3-free d-degenerate graphs and then derives the two extensions via independent entropy analysis of probability distributions on independent sets. These entropy calculations directly produce the stated O(d log(2r)/log d) and c_r (d/log d)^{1/(r-1)} bounds under the d-degeneracy and girth-at-least-4 assumptions without any fitted parameters, self-definitional reductions, or load-bearing self-citations. The derivation chain remains self-contained against external benchmarks and does not reduce by construction to its inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 3 axioms · 0 invented entities

The paper relies on standard definitions of degeneracy and girth together with entropy properties of discrete distributions; one constant c_r is introduced to absorb r-dependent factors.

free parameters (1)
  • c_r
    Constant depending on uniformity parameter r that absorbs lower-order factors in the hypergraph bound.
axioms (3)
  • standard math Entropy is subadditive and satisfies standard inequalities for discrete probability distributions
    Invoked in the analysis of the random distributions used to find independent sets.
  • domain assumption A d-degenerate graph has every subgraph with a vertex of degree at most d
    Standard definition used to set up the inductive or probabilistic argument.
  • domain assumption Girth at least 4 means no 2-cycles or 3-cycles in the hypergraph
    Used to control overlaps in the hypergraph setting.

pith-pipeline@v0.9.0 · 5597 in / 1463 out tokens · 63499 ms · 2026-05-15T08:49:30.210436+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Algorithmic Phase Transition for Large Independent Sets in Dense Hypergraphs

    cs.DS 2026-05 unverdicted novelty 7.0

    Online algorithms achieve multiplicative approximation r^{1/(r-1)} for maximum independent sets in dense r-uniform ER hypergraphs and (max γ_i)^{-1/(r-1)} for balanced sets in r-partite versions, with matching lower bounds.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Extremal uncrowded hypergraphs

    M. Ajtai, J. Koml´ os, J. Pintz, J. Spencer, and E. Szemer´ edi. “Extremal uncrowded hypergraphs”. In:Journal of Combinatorial Theory, Series A32.3 (1982), pp. 321–335 (cit. on pp. 2, 3)

    work page 1982

  2. [2]

    Coloring graphs with sparse neighborhoods

    N. Alon, M. Krivelevich, and B. Sudakov. “Coloring graphs with sparse neighborhoods”. In:J. Combin. Theory. B 77 (1999), pp. 73–82 (cit. on pp. 1, 3)

    work page 1999

  3. [3]

    Alon and J

    N. Alon and J. Spencer.The Probabilistic Method. 2nd ed. John Wiley & Sons, 2000 (cit. on p. 4)

    work page 2000

  4. [4]

    Independence numbers of locally sparse graphs and a Ramsey type problem

    N. Alon. “Independence numbers of locally sparse graphs and a Ramsey type problem”. In:Random Structures Algorithms9.3 (1996), pp. 271–278 (cit. on p. 2)

    work page 1996

  5. [5]

    Coloring graphs with forbidden bipartite subgraphs

    J. Anderson, A. Bernshteyn, and A. Dhawan. “Coloring graphs with forbidden bipartite subgraphs”. In:Combinatorics, Probability and Computing32.1 (2023), pp. 45–67 (cit. on p. 1)

    work page 2023

  6. [6]

    Coloring graphs with forbidden almost bipartite subgraphs

    J. Anderson, A. Bernshteyn, and A. Dhawan. “Coloring graphs with forbidden almost bipartite subgraphs”. In:Random Structures & Algorithms66.4 (2025), e70012 (cit. on p. 1)

    work page 2025

  7. [7]

    Coloring locally sparse graphs

    J. Anderson, A. Dhawan, and A. Kuchukova. “Coloring locally sparse graphs”. In:The Electronic Journal of Combinatorics(2026), P1.31 (cit. on p. 1)

    work page 2026

  8. [8]

    On the Lov\'asz Theta function for Independent Sets in Sparse Graphs

    N. Bansal, A. Gupta, and G. Guruganesh. “On the Lov´ asz theta function for independent sets in sparse graphs”. In:Proceedings of the forty-seventh annual ACM symposium on Theory of Com- puting. 2015, pp. 193–200. Full version:https://arxiv.org/abs/1504.04767(cit. on p. 6)

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  9. [9]

    The Johansson-Molloy theorem for DP-coloring

    A. Bernshteyn. “The Johansson-Molloy theorem for DP-coloring”. In:Random Structures & Algo- rithms54.4 (2019), pp. 653–664 (cit. on pp. 1, 6)

    work page 2019

  10. [10]

    Boundingχby a fraction of ∆ for graphs without large cliques

    M. Bonamy, T. Kelly, P. Nelson, and L. Postle. “Boundingχby a fraction of ∆ for graphs without large cliques”. In:Journal of Combinatorial Theory, Series B157 (2022), pp. 263–282 (cit. on pp. 1, 2, 6)

    work page 2022

  11. [11]

    Coloring small locally sparse degenerate graphs and related problems

    D. Bradaˇ c, J. Fox, R. Steiner, B. Sudakov, and S. Zhang. “Coloring small locally sparse degenerate graphs and related problems”.https://arxiv.org/abs/2601.15245(preprint). 2026 (cit. on pp. 1, 7)

    work page arXiv 2026

  12. [12]

    On colouring the nodes of a network

    R. L. Brooks. “On colouring the nodes of a network”. In:Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 37. 2. Cambridge University Press. 1941, pp. 194–197 (cit. on p. 1)

    work page 1941

  13. [13]

    List coloring triangle-free hypergraphs

    J. Cooper and D. Mubayi. “List coloring triangle-free hypergraphs”. In:Random Structures & Algorithms47.3 (2015), pp. 487–519 (cit. on p. 7)

    work page 2015

  14. [14]

    Graph structure via local occupancy

    E. Davies, R. Kang, F. Pirot, and J.-S. Sereni. “Graph structure via local occupancy”.https : //arxiv.org/abs/2003.14361(preprint). 2020 (cit. on pp. 1, 2, 6). 20

    work page arXiv 2003

  15. [15]

    On the average size of independent sets in triangle-free graphs

    E. Davies, M. Jenssen, W. Perkins, and B. Roberts. “On the average size of independent sets in triangle-free graphs”. In:Proceedings of the American Mathematical Society146.1 (2018), pp. 111– 124 (cit. on p. 1)

    work page 2018

  16. [16]

    Solution to advanced problem no. 4526

    B. Descartes. “Solution to advanced problem no. 4526”. In:Amer. Math. Monthly61.352 (1954), p. 216 (cit. on p. 1)

    work page 1954

  17. [17]

    Palette Sparsification for Graphs with Sparse Neighborhoods

    A. Dhawan. “Palette Sparsification for Graphs with Sparse Neighborhoods”.https://arxiv.org/ abs/2408.08256(preprint). 2024 (cit. on p. 1)

    work page arXiv 2024

  18. [18]

    Bounds for the Independence and Chromatic Numbers of Locally Sparse Graphs

    A. Dhawan. “Bounds for the Independence and Chromatic Numbers of Locally Sparse Graphs”. In: Annals of Combinatorics(2025), pp. 1–28 (cit. on pp. 1, 6)

    work page 2025

  19. [19]

    Independent sets and colorings ofK t,t,t-free graphs

    A. Dhawan, O. Janzer, and A. Methuku. “Independent sets and colorings ofK t,t,t-free graphs”. https://arxiv.org/abs/2511.17191(preprint). 2025 (cit. on pp. 1, 3)

    work page arXiv 2025

  20. [20]

    Improved bounds on the independence number of un- crowded hypergraphs and triangle-free hypergraphs

    A. Dhawan, A. Methuku, and M.-Q. Vo. “Improved bounds on the independence number of un- crowded hypergraphs and triangle-free hypergraphs”. In preparation (cit. on p. 3)

    work page

  21. [21]

    On uncrowded hypergraphs

    R. A. Duke, H. Lefmann, and V. R¨ odl. “On uncrowded hypergraphs”. In:Random Structures & Algorithms6.2-3 (1995), pp. 209–212 (cit. on p. 3)

    work page 1995

  22. [22]

    Coloring simple hypergraphs

    A. Frieze and D. Mubayi. “Coloring simple hypergraphs”. In:Journal of Combinatorial Theory, Series B103.6 (2013), pp. 767–794 (cit. on pp. 3, 5, 7)

    work page 2013

  23. [23]

    On the independence number of random graphs

    A. M. Frieze. “On the independence number of random graphs”. In:Discrete Mathematics81.2 (1990), pp. 171–175 (cit. on p. 2)

    work page 1990

  24. [24]

    Some results on chromatic number as a function of triangle count

    D. G. Harris. “Some results on chromatic number as a function of triangle count”. In:SIAM Journal on Discrete Mathematics33.1 (2019), pp. 546–563 (cit. on p. 1)

    work page 2019

  25. [25]

    Improved Bounds for Coloring Locally Sparse Hypergraphs

    F. Iliopoulos. “Improved Bounds for Coloring Locally Sparse Hypergraphs”. In:Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques(2021) (cit. on p. 3)

    work page 2021

  26. [26]

    Johansson.Asymptotic choice number for triangle free graphs

    A. Johansson.Asymptotic choice number for triangle free graphs. Technical Report 91–95. DIMACS, 1996 (cit. on pp. 1, 6)

    work page 1996

  27. [27]

    The choice number of sparse graphs

    A. Johansson. “The choice number of sparse graphs”. Unpublished Manuscript. 1996 (cit. on p. 6)

    work page 1996

  28. [28]

    Graph and hypergraph colouring via nibble methods: A survey

    D. Y. Kang, T. Kelly, D. K¨ uhn, A. Methuku, and D. Osthus. “Graph and hypergraph colouring via nibble methods: A survey”. In:Proceedings of the 8th European Congress of Mathematics. EMS Press, 2023, pp. 771–823.doi:10.4171/8ECM/11(cit. on p. 6)

    work page doi:10.4171/8ecm/11(cit 2023

  29. [29]

    Fractional coloring with local demands and applications to degree-sequence bounds on the independence number

    T. Kelly and L. Postle. “Fractional coloring with local demands and applications to degree-sequence bounds on the independence number”. In:Journal of Combinatorial Theory, Series B169 (2024), pp. 298–337 (cit. on p. 1)

    work page 2024

  30. [30]

    Properties of Descartes’ construction of triangle-free graphs with high chromatic number

    A. V. Kostochka and J. Neˇ setˇ ril. “Properties of Descartes’ construction of triangle-free graphs with high chromatic number”. In:Combinatorics, Probability and Computing8.5 (1999), pp. 467–472 (cit. on p. 1)

    work page 1999

  31. [31]

    The chromatic number of triangle-free hypergraphs

    L. Li and L. Postle. “The chromatic number of triangle-free hypergraphs”.https://arxiv.org/ abs/2202.02839(preprint). 2022 (cit. on pp. 3, 7)

    work page arXiv 2022

  32. [32]

    Random independent sets in triangle-free graphs

    A. Martinsson and R. Steiner. “Random independent sets in triangle-free graphs”. In:Forum of Mathematics, Sigma. Vol. 13. Cambridge University Press. 2025, e156 (cit. on pp. 1, 3)

    work page 2025

  33. [33]

    The list chromatic number of graphs with small clique number

    M. Molloy. “The list chromatic number of graphs with small clique number”. In:J. Combin. Theory. B 134 (2019), pp. 264–284 (cit. on pp. 1, 6)

    work page 2019

  34. [34]

    Springer Dordrecht, 1988.doi:10.1007/978- 94-009-2871-8

    M. Molloy and B. Reed.Graph Colouring and the Probabilistic Method. Vol. 23. Algorithms and Combinatorics. Springer-Verlag, Berlin, 2002, pp. xiv+326.isbn: 3-540-42139-4.doi:10.1007/978- 3-642-04016-0.url:https://doi.org/10.1007/978-3-642-04016-0(cit. on p. 4)

    work page doi:10.1007/978- 2002

  35. [35]

    Distributed coloring algorithms for triangle-free graphs

    S. Pettie and H.-H. Su. “Distributed coloring algorithms for triangle-free graphs”. In:Information and Computation243 (2015), pp. 263–280 (cit. on p. 1)

    work page 2015

  36. [36]

    On the independence number of sparse graphs

    J. B. Shearer. “On the independence number of sparse graphs”. In:Random Structures & Algorithms 7.3 (1995), pp. 269–271 (cit. on p. 6). 21

    work page 1995

  37. [37]

    Independent sets in hypergraphs

    J. Verstraete and C. Wilson. “Independent sets in hypergraphs”. In:Random Structures & Algo- rithms68.1 (2026), e70047 (cit. on p. 3)

    work page 2026