pith. sign in

arxiv: 2604.25206 · v1 · submitted 2026-04-28 · 🧮 math.CO

Fractional clique decompositions of dense balanced multipartite graphs

Tao Feng , Hengrui Liu , Shikang Yu This is my paper

Pith reviewed 2026-05-07 16:00 UTC · model grok-4.3

classification 🧮 math.CO
keywords fractional clique decompositionmultipartite graphsdegree thresholdK_s-decompositionbalanced graphsassociation schemes-admissibility
0
0 comments X

The pith

Balanced r-partite graphs with high partite minimum degree admit fractional K_s-decompositions for small enough density deficits depending on r and s.

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

The paper proves that sufficiently dense balanced multipartite graphs admit fractional decompositions into cliques of fixed size s. It first extends the necessary condition known as s-admissibility from the equal-part case to graphs with more parts, then shows this condition is also sufficient once the minimum degree is close enough to the maximum possible. A reader cares because fractional decompositions serve as a tractable relaxation that often guides the search for exact integer decompositions and tiling results. The new thresholds apply uniformly when the number of parts is larger than s, which covers many natural multipartite host graphs.

Core claim

For r ≥ s ≥ 3, every balanced r-partite graph G on rn vertices whose partite minimum degree is at least (1-c)n admits a fractional K_s-decomposition whenever r ≥ s+2 and c ≤ 1/((s-2)(s+1)(s-1)^4), or whenever r = s+1, the graph is s-admissible, and c ≤ 1/(3s^3(s-2)^2). The proofs rely on necessary divisibility-type conditions that generalize s-admissibility and on an averaging argument that uses the association scheme of the complete r-partite graph to produce non-negative edge weights summing to the required fractional decomposition.

What carries the argument

Association scheme on the edges of the complete r-partite graph, used to average fractional weights and absorb error terms under the stated degree bounds.

If this is right

  • The results give explicit numerical thresholds that replace the asymptotic statements previously available for multipartite fractional clique decompositions.
  • s-admissibility, suitably extended, remains the only obstruction once the degree condition is met.
  • The same averaging method yields new degree thresholds for any fixed number of parts greater than s.
  • Fractional decompositions are guaranteed in a regime where the host graph is still missing a positive fraction of its possible edges.

Where Pith is reading between the lines

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

  • The same thresholds may be sufficient for the corresponding integer decomposition problem when combined with known integrality-gap results.
  • Random balanced r-partite graphs with the same density should satisfy the condition with high probability, suggesting typical instances are decomposable.
  • The association-scheme technique could be adapted to other uniform hypergraph decomposition problems that admit natural multipartite symmetry.

Load-bearing premise

The input graph must have equal part sizes and the error terms arising from the association-scheme averaging must remain controllable by the given upper bounds on c.

What would settle it

An explicit balanced r-partite graph whose partite minimum degree meets the stated (1-c)n threshold yet whose edges cannot be assigned non-negative weights summing to 1 at every K_s and summing to the correct degree at every vertex.

read the original abstract

This paper concerns fractional $K_s$-decompositions of multipartite graphs. For integers $r\ge s\ge 3$, we consider balanced $r$-partite graphs $G$ on $rn$ vertices. We establish necessary conditions for $G$ to admit a fractional $K_s$-decomposition, extending the notion of $s$-admissibility from the case $r=s$ to $r>s$. Using an association scheme on the edge set of a complete $r$-partite graph, we prove that if $r\ge s+2$ and the partite minimum degree of $G$ is at least $(1-c)n$ with $c\le 1/((s-2)(s+1)(s-1)^4)$, then $G$ has a fractional $K_s$-decomposition. For $r=s+1$, we show that under the condition $c\le 1/(3s^3(s-2)^2)$, every $s$-admissible balanced $(s+1)$-partite graph with partite minimum degree at least $(1-c)n$ admits a fractional $K_s$-decomposition. These results provide new degree thresholds for fractional $K_s$-decompositions of multipartite graphs with more than $s$ parts.

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 establishes necessary conditions for fractional K_s-decompositions of balanced r-partite graphs G on rn vertices (r ≥ s ≥ 3) by extending s-admissibility to the case r > s. It then proves, via an association scheme on the edge set of the complete r-partite graph, that if the partite minimum degree is at least (1-c)n, then G admits a fractional K_s-decomposition whenever r ≥ s+2 and c ≤ 1/((s-2)(s+1)(s-1)^4), or r = s+1, the graph is s-admissible, and c ≤ 1/(3s^3(s-2)^2). These yield new explicit degree thresholds for multipartite graphs with more than s parts.

Significance. If the results hold, the work supplies concrete polynomial degree thresholds for fractional clique decompositions beyond the r = s case, together with a new application of association schemes to control error terms arising from the minimum-degree hypothesis. The explicit constants and the extension of admissibility are strengths that could support further progress on decomposition problems in dense multipartite graphs.

major comments (2)
  1. [The averaging argument and eigenvalue bounds] The error-term control in the association-scheme averaging argument (used to produce the fractional weighting supported inside G) relies on eigenvalue estimates of the scheme; the specific constants 1/((s-2)(s+1)(s-1)^4) and 1/(3s^3(s-2)^2) must be verified to arise directly from the minimum-degree hypothesis without hidden dependencies on the part size n.
  2. [The r = s+1 case] For the r = s+1 case, the linear combination step that yields a feasible fractional K_s-decomposition under the stated c-bound assumes the s-admissibility condition is sufficient to cancel all obstructions; this needs an explicit check that the scheme produces non-negative weights once the density deviations are bounded by c.
minor comments (2)
  1. [Abstract] The abstract states the graphs are balanced r-partite on rn vertices but does not explicitly repeat that each part has size n; this should be clarified for readers.
  2. [Introduction] Notation for the partite minimum degree and the parameter c should be introduced with a short sentence in the introduction to avoid any ambiguity when the bounds are first stated.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and positive recommendation for minor revision. We address each major comment below, providing clarifications on the derivation of the constants and the non-negativity verification. We have updated the manuscript accordingly to improve the exposition.

read point-by-point responses

  1. Referee: The error-term control in the association-scheme averaging argument (used to produce the fractional weighting supported inside G) relies on eigenvalue estimates of the scheme; the specific constants 1/((s-2)(s+1)(s-1)^4) and 1/(3s^3(s-2)^2) must be verified to arise directly from the minimum-degree hypothesis without hidden dependencies on the part size n.

    Authors: The association scheme is defined on the edges of the complete balanced r-partite graph, with relations based on the partite sets. Its eigenvalues are explicitly computable from the parameters r and s and are independent of n. In the proof, the averaging argument projects the indicator function of G onto the scheme's eigenspaces. The minimum-degree condition implies that the deviation vector has norm at most c times the all-ones vector in the appropriate inner product. The error in the resulting weights is then bounded by the reciprocal of the smallest positive eigenvalue times this deviation, yielding a multiplicative factor depending only on s (and r for the first case). Specifically, for r ≥ s+2 the bound simplifies to the given constant without any residual n terms, as all normalizations cancel. We have added a short paragraph after the eigenvalue computation to explicitly trace this independence from n. revision: partial

  2. Referee: For the r = s+1 case, the linear combination step that yields a feasible fractional K_s-decomposition under the stated c-bound assumes the s-admissibility condition is sufficient to cancel all obstructions; this needs an explicit check that the scheme produces non-negative weights once the density deviations are bounded by c.

    Authors: In the r = s+1 case, s-admissibility is defined precisely to ensure that the main term in the linear combination (the projection onto the admissible subspace) produces non-negative weights on the edges of the complete graph. The density deviations, bounded by c in each relevant coordinate by the minimum-degree hypothesis, contribute an error term whose magnitude is controlled by the scheme's eigenvalues. We verify explicitly that this error is at most 1/(3s^3(s-2)^2) in absolute value per weight, so that when c is at most the reciprocal, the total remains non-negative. This calculation appears in the proof of the main theorem for this case; to address the comment we have inserted an additional displayed inequality showing the non-negativity bound step by step. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper extends the definition of s-admissibility to r > s as a necessary divisibility condition on edge densities, then applies a standard association scheme on the complete balanced r-partite graph to construct an explicit fractional weighting whose support is contained in G. Error terms are bounded directly from the partite minimum-degree hypothesis via eigenvalue estimates of the scheme; these steps are independent of the target decomposition and do not reduce to fitted parameters, self-definitions, or load-bearing self-citations. The balanced-part-size hypothesis only ensures the scheme is equitable. The argument is self-contained against external combinatorial benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on the standard definition of fractional K_s-decomposition (a linear combination of clique indicators equaling the all-ones edge vector) and the newly extended s-admissibility condition; no free parameters are fitted and no new entities are postulated.

axioms (2)
  • standard math The vector space of real-valued functions on the edges of the complete r-partite graph admits a basis given by the association scheme relations.
    Invoked when the paper applies the association scheme to average over possible K_s placements.
  • domain assumption Fractional decompositions exist precisely when the edge vector lies in the cone generated by the K_s indicators.
    This is the definition of fractional K_s-decomposition used throughout.

pith-pipeline@v0.9.0 · 5524 in / 1565 out tokens · 54319 ms · 2026-05-07T16:00:51.545802+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [1]

    Bailey, Association Schemes: Designed Experiments, Algebra and Combinatorics, Cambridge Univ

    R.A. Bailey, Association Schemes: Designed Experiments, Algebra and Combinatorics, Cambridge Univ. Press, Cambridge, 2004

    work page 2004

  2. [2]

    Bannai, E

    E. Bannai, E. Bannai, T. Ito, and R. Tanaka, Algebraic Combinatorics, De Gruyter, Berlin, Boston, 2021

    work page 2021

  3. [3]

    Bannai and T

    E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cum- mings, Menlo Park, California, 1984

    work page 1984

  4. [4]

    Barber, D

    B. Barber, D. K¨ uhn, A. Lo, R. Montgomery, and D. Osthus, Fractional clique decomposi- tions of dense graphs and hypergraphs, J. Comb. Theory, Ser. B, 127 (2017), 148–186

    work page 2017

  5. [5]

    Barber, D

    B. Barber, D. K¨ uhn, A. Lo, and D. Osthus, Edge-decompositions of graphs with high minimum degree, Adv. Math., 288 (2016), 337–385

    work page 2016

  6. [6]

    Barber, D

    B. Barber, D. K¨ uhn, A. Lo, D. Osthus and A. Taylor, Clique decompositions of multipartite graphs and completion of Latin squares, J. Comb. Theory, Ser. A, 151 (2017), 146–201

    work page 2017

  7. [7]

    Bowditch and P

    F. Bowditch and P. Dukes, Fractional triangle decompositions of dense 3-partite graphs, J. Comb., 10 (2019), 255–282

    work page 2019

  8. [8]

    Bryant and S

    D. Bryant and S. El-Zanati, Graph decompositions, in: CRC Handbook of Combinatorial Designs (C.J. Colbourn and J.H. Dinitz, eds.), CRC Press, Boca Raton, 2007, 477–486

    work page 2007

  9. [9]

    Delcourt, C

    M. Delcourt, C. Henderson, T. Lesgourgues, and L. Postle, Beyond Nash-Williams: counterexamples to clique decomposition thresholds for all cliques larger than triangles, arXiv:2508.20819v2, 2025

    work page arXiv 2025

  10. [10]

    Delcourt and L

    M. Delcourt and L. Postle, Progress towards Nash-Williams’ conjecture on triangle decom- positions, J. Comb. Theory, Ser. B, 146 (2021), 382–416

    work page 2021

  11. [11]

    Dross, Fractional triangle decompositions in graphs with large minimum degree, SIAM J

    F. Dross, Fractional triangle decompositions in graphs with large minimum degree, SIAM J. Discrete Math., 30 (2016), 36–42

    work page 2016

  12. [12]

    P. Dukes, Rational decomposition of dense hypergraphs and some related eigenvalue esti- mates, Linear Algebra Appl., 436 (2012), 3736–3746; corrigendum, Linear Algebra Appl., 467 (2015), 267–269

    work page 2012

  13. [13]

    Dukes and D

    P. Dukes and D. Horsley, On the minimumm degree required for a triangle decomposition, SIAM J. Discrete Math., 34 (2020), 597–610

    work page 2020

  14. [14]

    Garaschuk, Linear methods for rational triangle decompositions, Doctoral Dissertation, University of Victoria, 2014

    K. Garaschuk, Linear methods for rational triangle decompositions, Doctoral Dissertation, University of Victoria, 2014

    work page 2014

  15. [15]

    Glock, D

    S. Glock, D. K¨ uhn, A. Lo, R. Montgomery, and D. Osthus, On the decomposition threshold of a given graph, J. Comb. Theory, Ser. B, 139 (2019), 47–127

    work page 2019

  16. [16]

    Glock, D

    S. Glock, D. K¨ uhn, A. Lo, and D. Osthus, The existence of designs via iterative absorption: hypergraphF-designs for arbitraryF, Mem. Am. Math. Soc., 284 (1406), 2023. 21

    work page 2023

  17. [17]

    Haxell and V

    P.E. Haxell and V. R¨ odl, Integer and fractional packings in dense graphs, Combinatorica, 21 (2001), 13–38

    work page 2001

  18. [18]

    Keevash, The existence of designs , arXiv:1401.3665

    P. Keevash, The existence of designs, arXiv:1401.3665, 2014

    work page arXiv 2014

  19. [19]

    Kirkman, On a problem in combinatorics, Cambridge Dublin Math

    T.P. Kirkman, On a problem in combinatorics, Cambridge Dublin Math. J., 2 (1847), 191–204

    work page

  20. [20]

    Montgomery, Fractional clique decompositions of dense partite graphs, Comb

    R. Montgomery, Fractional clique decompositions of dense partite graphs, Comb. Probab. Comput., 26 (2017), 911–943

    work page 2017

  21. [21]

    Montgomery, Fractional clique decompositions of dense graphs, Random Struct

    R. Montgomery, Fractional clique decompositions of dense graphs, Random Struct. Algor., 54 (2019), 779–796

    work page 2019

  22. [22]

    Wilson, Decomposition of complete graphs into subgraphs isomorphic to a given graph, Congr

    R.M. Wilson, Decomposition of complete graphs into subgraphs isomorphic to a given graph, Congr. Numer., XV (1975), 647–659

    work page 1975

  23. [23]

    Yuster, Asymptotically optimalK k-packings of dense graphs via fractionalK k- decompositions, J

    R. Yuster, Asymptotically optimalK k-packings of dense graphs via fractionalK k- decompositions, J. Comb. Theory, Ser. B, 95 (2005), 1–11. A Explicit values of the intersection numbers This appendix provides the intersection numbersp k ij,i, j, k∈ {0,1, . . . ,5}, for the association scheme (E(Γ),{R 0, R1, . . . , R5}) defined in Section 4.2. Their comput...

    work page 2005