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arxiv: 2604.25206 · v2 · pith:DAKX4PHTnew · 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 2026-07-01 09:24 UTC · model grok-4.3

classification 🧮 math.CO
keywords fractional clique decompositionmultipartite graphsminimum degrees-admissibilityassociation schemebalanced graphsK_s-decomposition
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The pith

Balanced r-partite graphs admit fractional K_s-decompositions when partite minimum degree reaches (1-c)n for small c depending on s.

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

The paper establishes necessary conditions for a balanced r-partite graph on rn vertices to admit a fractional K_s-decomposition by extending the notion of s-admissibility from the equal-part case r equals s to larger r. It then proves sufficiency of high minimum degree: for r at least s plus 2 the degree bound alone works when c is at most 1 over ((s-2)(s+1)(s-1)^4), while for r equals s plus 1 the same conclusion holds for every s-admissible graph when c is at most 1 over (3s cubed times (s-2) squared). These statements are proved via an association scheme on the edges of the complete r-partite graph. A reader would care because the results supply explicit degree thresholds that apply once the number of parts exceeds s.

Core claim

For integers r greater than or equal to s greater than or equal to 3, if r is at least s plus 2 and the partite minimum degree of G is at least (1-c)n with c at most 1/((s-2)(s+1)(s-1)^4), then G has a fractional K_s-decomposition. For r equals s plus 1, every s-admissible balanced (s+1)-partite graph with partite minimum degree at least (1-c)n where c is at most 1/(3s^3(s-2)^2) admits a fractional K_s-decomposition. The proofs first derive the necessary s-admissibility conditions and then use an association scheme to show the degree conditions suffice.

What carries the argument

The association scheme on the edge set of the complete r-partite graph, which encodes the edge types and is used to verify that the degree conditions imply the existence of a fractional K_s-decomposition.

If this is right

  • For r at least s plus 2 the s-admissibility check is unnecessary once the minimum-degree bound holds.
  • Explicit quantitative degree thresholds are now available for fractional K_s-decompositions in multipartite graphs with any number of parts at least s plus 1.
  • The same association-scheme method yields thresholds whose dependence on s is independent of r when r exceeds s plus 1.
  • The results extend earlier degree conditions known only for the balanced complete s-partite case to graphs with extra parts.

Where Pith is reading between the lines

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

  • The fractional thresholds may serve as a stepping stone toward integer K_s-decompositions if an integrality gap can be controlled separately.
  • The association-scheme technique could be adapted to obtain analogous results for other subgraph decompositions such as fractional designs in multipartite host graphs.
  • Improving the explicit constants on c would require only a tighter eigenvalue or linear-programming analysis of the same scheme.

Load-bearing premise

The extension of the s-admissibility condition from the r equals s case to r greater than s is correctly formulated as a necessary condition that remains compatible with the association scheme analysis used to prove sufficiency under the stated degree bounds.

What would settle it

A concrete balanced r-partite graph on rn vertices whose partite minimum degree meets the stated (1-c)n threshold yet possesses no fractional K_s-decomposition (or, for the r equals s plus 1 case, an s-admissible graph meeting the bound that still lacks the decomposition).

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

3 major / 2 minor

Summary. The paper claims that for integers r ≥ s ≥ 3, balanced r-partite graphs G on rn vertices admit fractional K_s-decompositions under explicit partite minimum-degree thresholds. It extends the notion of s-admissibility from the r = s case to r > s as a necessary condition. Using an association scheme on the edge set of the complete r-partite graph K_r(n), it proves that if r ≥ s+2 and the partite minimum degree is at least (1-c)n with c ≤ 1/((s-2)(s+1)(s-1)^4), then G has a fractional K_s-decomposition. For r = s+1, under the stricter bound c ≤ 1/(3s^3(s-2)^2), every s-admissible such graph admits the decomposition. These provide new degree thresholds for multipartite graphs with more than s parts.

Significance. If the results hold, they extend known degree conditions for fractional clique decompositions to multipartite graphs with r > s parts via algebraic methods (association schemes and the Bose-Mesner algebra), providing explicit constants c that are load-bearing for the sufficiency claims. The extension of admissibility and the handling of orbit subsets for K_s when r > s would be a technical contribution if the compatibility is rigorously verified.

major comments (3)
  1. [necessary conditions section] The extension of s-admissibility to r > s (section on necessary conditions): the abstract states that this extension remains compatible with the association scheme analysis, but when s < r the K_s-subgraphs span only a subset of the r-partite edge orbits. The paper must explicitly show that the non-negativity and integrality conditions on the fractional weights in the LP (solved via the idempotents of the Bose-Mesner algebra) are implied by the minimum-degree hypothesis alone for r ≥ s+2, without hidden linear dependencies.
  2. [main theorem for r ≥ s+2] Proof of sufficiency for r ≥ s+2 (main theorem): the association scheme is defined on the full set of edge orbits of K_r(n), yet the fractional decomposition is over K_s copies. It is unclear how the projection onto the relevant subalgebra ensures that the degree condition (1-c)n with the stated c suffices to guarantee a non-negative solution without additional constraints; this is load-bearing for the claim that admissibility is unnecessary in this regime.
  3. [proofs of the degree thresholds] The specific numerical bounds on c (e.g., 1/((s-2)(s+1)(s-1)^4) for r ≥ s+2 and 1/(3s^3(s-2)^2) for r = s+1): these appear derived from the eigenvalues or intersection numbers of the scheme, but the manuscript should verify that they are not post-hoc and that smaller c would fail for some constructions.
minor comments (2)
  1. [necessary conditions] Clarify the precise definition of the extended s-admissibility condition with an explicit formula or set of linear equations it imposes on the degree sequence.
  2. [introduction] Add a short comparison table or paragraph contrasting the new c bounds with prior thresholds from the r = s case.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address each of the major comments below, providing clarifications and indicating the revisions we will make to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [necessary conditions section] The extension of s-admissibility to r > s (section on necessary conditions): the abstract states that this extension remains compatible with the association scheme analysis, but when s < r the K_s-subgraphs span only a subset of the r-partite edge orbits. The paper must explicitly show that the non-negativity and integrality conditions on the fractional weights in the LP (solved via the idempotents of the Bose-Mesner algebra) are implied by the minimum-degree hypothesis alone for r ≥ s+2, without hidden linear dependencies.

    Authors: We agree that an explicit demonstration is necessary to confirm the absence of hidden linear dependencies. In the revised version, we will add a new subsection in the necessary conditions part that projects the admissibility conditions onto the subalgebra generated by the K_s orbits and verifies that the minimum-degree condition directly implies the non-negativity of the fractional weights via the idempotent basis, without requiring additional constraints for r ≥ s+2. revision: yes

  2. Referee: [main theorem for r ≥ s+2] Proof of sufficiency for r ≥ s+2 (main theorem): the association scheme is defined on the full set of edge orbits of K_r(n), yet the fractional decomposition is over K_s copies. It is unclear how the projection onto the relevant subalgebra ensures that the degree condition (1-c)n with the stated c suffices to guarantee a non-negative solution without additional constraints; this is load-bearing for the claim that admissibility is unnecessary in this regime.

    Authors: To clarify this point, we will insert a dedicated lemma following the definition of the association scheme. This lemma will explicitly describe the projection map onto the subalgebra corresponding to the K_s edge orbits and show, using the given bound on c, that the resulting linear system has a non-negative solution solely from the partite minimum-degree assumption. This will make the independence from admissibility rigorous. revision: yes

  3. Referee: [proofs of the degree thresholds] The specific numerical bounds on c (e.g., 1/((s-2)(s+1)(s-1)^4) for r ≥ s+2 and 1/(3s^3(s-2)^2) for r = s+1): these appear derived from the eigenvalues or intersection numbers of the scheme, but the manuscript should verify that they are not post-hoc and that smaller c would fail for some constructions.

    Authors: The constants arise from solving the inequalities required for positivity in the closed-form solution obtained from the Bose-Mesner algebra eigenvalues. We will expand the proof section with a step-by-step computation of these bounds from the intersection numbers to demonstrate they follow directly from the analysis. However, we note that establishing that smaller values of c fail would necessitate explicit constructions of graphs violating the decomposition, which lies beyond the scope of the current sufficient-condition results. revision: partial

standing simulated objections not resolved
  • Providing constructions to show that the given bounds on c are sharp, as this would require additional extremal graph theory results not developed in the paper.

Circularity Check

0 steps flagged

No significant circularity; direct proof via association schemes

full rationale

The derivation establishes necessary conditions by extending s-admissibility (defined explicitly for r>s) and then applies the Bose-Mesner algebra of the standard r-partite association scheme to obtain non-negative solutions to the fractional decomposition LP under the stated minimum-degree bounds. No step reduces a claimed prediction or theorem to a fitted parameter, self-citation chain, or input by construction; the scheme is invoked as an external algebraic tool whose idempotents yield the required weights independently of the target result. The r=s+1 case adds the admissibility hypothesis explicitly rather than deriving it from the degree condition alone.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on the abstract alone, the work extends existing combinatorial notions without introducing new free parameters or postulated entities. It relies on standard properties of association schemes and the definition of fractional decompositions.

axioms (2)
  • standard math Standard properties of association schemes on the edges of complete multipartite graphs
    Invoked to organize edge types and prove existence of non-negative fractional weights.
  • domain assumption The extended definition of s-admissibility for r > s is a valid necessary condition
    Stated as established in the paper before sufficiency is proved under degree conditions.

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Forward citations

Cited by 1 Pith paper

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

  1. On the completion of $\epsilon$-dense partial Latin squares

    math.CO 2026-06 unverdicted novelty 7.0

    Every sufficiently large 2/25-dense partial Latin square is completable, established through a fractional triangle decomposition result for balanced tripartite graphs.

Reference graph

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23 extracted references · 2 canonical work pages · cited by 1 Pith paper

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