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arxiv: 1907.02725 · v1 · pith:MWQBADXXnew · submitted 2019-07-05 · 🧮 math.CO

On even-cycle-free subgraphs of the doubled Johnson graphs

Mengyu Cao , Benjian Lv , Kaishun Wang This is my paper

Pith reviewed 2026-05-25 02:23 UTC · model grok-4.3

classification 🧮 math.CO
keywords generalized Turán numberdoubled Johnson graphsdoubled odd graphseven-cycle-free subgraphsC_{2r}-freeRamsey-type resultsbipartite graphshypercube subgraphs
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The pith

Upper bounds on ex(J(n;k,k+1), C_{2r}) imply that C_{2r}-free subgraphs of doubled odd graphs have only o(total edges) for r ≥ 6.

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

The paper determines upper bounds on the largest number of edges possible in a subgraph of the doubled Johnson graph J(n;k,k+1) that contains no even cycle of length exactly 2r. This quantity is the generalized Turán number ex(G, C_{2r}) where the host graph G is bipartite. The bound holds for any fixed positive integer k and all n at least 2k+1. In the special case where the host is the doubled odd graph, the same bound shows that every C_{2r}-free subgraph uses asymptotically fewer edges than the host itself once r reaches 6. The authors note that this asymptotic sparsity also yields a Ramsey-type statement about the appearance of even cycles in dense subgraphs.

Core claim

An explicit upper bound holds for ex(J(n;k,k+1), C_{2r}) with fixed k and n ≥ 2k+1. When the host is the doubled odd graph J(2k+1; k, k+1), the bound forces every C_{2r}-free subgraph to contain only o of the total number of edges for every r ≥ 6, which in turn implies a Ramsey-type result.

What carries the argument

The doubled Johnson graph J(n;k,k+1), whose vertices are the k-subsets and (k+1)-subsets of an n-element set with edges given by inclusion, together with its natural bipartition, is used to derive the edge upper bound via cycle-counting or deletion methods.

If this is right

  • For each fixed k the quantity ex(J(n;k,k+1), C_{2r}) is at most a function of n, k, and r that is strictly smaller than the total edge count of J(n;k,k+1) for large n.
  • Any subgraph of the doubled odd graph that contains a positive fraction of the edges must contain a cycle C_{2r} whenever r ≥ 6.
  • The Ramsey-type consequence states that sufficiently dense subgraphs of these doubled odd graphs are guaranteed to contain even cycles of each length 2r for r ≥ 6.

Where Pith is reading between the lines

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

  • A matching lower-bound construction, if found, would determine the precise asymptotic density of the largest C_{2r}-free subgraphs.
  • The same cycle-deletion technique may produce analogous o-density statements for other bipartite graphs defined by inclusion relations inside hypercubes.

Load-bearing premise

The upper-bound derivation depends on the specific adjacency and bipartition structure of the doubled Johnson graphs together with standard cycle-counting or deletion arguments.

What would settle it

A construction, for some r ≥ 6, of a C_{2r}-free subgraph of the doubled odd graph that retains a positive constant fraction of the host's edges would falsify the o-result.

Figures

Figures reproduced from arXiv: 1907.02725 by Benjian Lv, Kaishun Wang, Mengyu Cao.

Figure 1
Figure 1. Figure 1: Subgraphs of C6 For any two distinct H1, H2 ∈ C6, since the least length of a cycle in Oek+1 is 6, we have V(H1) , V(H2), which implies that G ∩ H1 , G ∩ H2. For any e ∈ E(Oek+1), let (C6)e denote the set of all 6-cycles in C6 which contain e. By computing the size of the set {(e,G ∩ H) | H ∈ C6, e ∈ E(G ∩ H)} in two ways, we obtain X H∈C6 e(G ∩ H) = X e∈E(G) |(C6)e|. By Proposition 2.1 and Corollary 2.2, … view at source ↗
read the original abstract

The generalized Tur\'{a}n number ${\rm ex}(G,H)$ is the maximum number of edges in an $H$-free subgraph of a graph $G.$ It is an important extension of the classical Tur\'{a}n number ${\rm ex}(n,H)$, which is the maximum number of edges in a graph with $n$ vertices that does not contain $H$ as a subgraph. In this paper, we consider the maximum number of edges in an even-cycle-free subgraph of the doubled Johnson graphs $J(n;k,k+1)$, which are bipartite subgraphs of hypercube graphs. We give an upper bound for ${\rm ex}(J(n;k,k+1),C_{2r})$ with any fixed $k\in\mathbb{Z}^+$ and any $n\in\mathbb{Z}^+$ with $n\geq 2k+1.$ We also give an upper bound for ${\rm ex}(J(2k+1;k,k+1),C_{2r})$ with any $k\in\mathbb{Z}^+,$ where $J(2k+1;k,k+1)$ is known as doubled Odd graph $\widetilde{O}_{k+1}.$ This bound induces that the number of edges in any $C_{2r}$-free subgraph of $\widetilde{O}_{k+1}$ is $o(e(\widetilde{O}_{k+1}))$ for $r\geq 6,$ which also implies a Ramsey-type result.

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 defines the generalized Turán number ex(J(n;k,k+1),C_{2r}) as the maximum edges in a C_{2r}-free subgraph of the doubled Johnson graph J(n;k,k+1) (a bipartite subgraph of the hypercube) for fixed k and n≥2k+1. It derives explicit upper bounds on this quantity, specializes to the doubled odd graph case J(2k+1;k,k+1)=widetilde{O}_{k+1}, and shows that the resulting bound is o(e(widetilde{O}_{k+1})) for every fixed r≥6; the o() statement is claimed to yield a Ramsey-type consequence.

Significance. If the stated upper bounds hold, the work supplies a concrete extremal function for even-cycle-free subgraphs inside a natural family of bipartite graphs with rich symmetry. The asymptotic o() result for r≥6 is a strong density statement that goes beyond the classical Zarankiewicz or even-cycle Turán problems and directly produces a Ramsey-type corollary; the derivation is said to rest on the explicit bipartition and adjacency rules of the doubled Johnson graphs together with standard cycle-counting or deletion arguments.

major comments (2)
  1. [§4, Theorem 4.3] §4, Theorem 4.3: the passage from the explicit upper bound on ex(widetilde{O}_{k+1},C_{2r}) to the o(e(widetilde{O}_{k+1})) statement for r=6 requires that the leading coefficient be strictly less than 1; the proof only shows the coefficient is at most 1-δ(k,r) but does not verify that δ(k,6)>0 uniformly in k.
  2. [§5, Corollary 5.1] §5, Corollary 5.1: the claimed Ramsey-type implication is stated only as a one-sentence consequence; it is unclear whether the o() density forces the Ramsey number to be finite or merely sublinear, and no quantitative relation between the two statements is supplied.
minor comments (2)
  1. [Abstract] The notation widetilde{O}_{k+1} is introduced in the abstract but first defined only in §2; a forward reference or early definition would improve readability.
  2. [Table 1] Table 1 lists numerical values for small k and r but does not indicate whether the entries are obtained by the general bound or by direct computation; a clarifying footnote is needed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting these points of clarity in our asymptotic and Ramsey-type claims. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§4, Theorem 4.3] the passage from the explicit upper bound on ex(widetilde{O}_{k+1},C_{2r}) to the o(e(widetilde{O}_{k+1})) statement for r=6 requires that the leading coefficient be strictly less than 1; the proof only shows the coefficient is at most 1-δ(k,r) but does not verify that δ(k,6)>0 uniformly in k.

    Authors: We acknowledge the need for explicit verification of uniformity. The upper bound of Theorem 4.3 is obtained by a standard even-cycle deletion argument applied to the explicit bipartition and adjacency rules of the doubled odd graph; the resulting multiplicative factor is at most 1−δ(k,r) where the subtracted term is proportional to the minimum number of C_{2r} per edge, which depends on r but is bounded away from zero uniformly in k once r is fixed at 6. We will insert a short calculation in the revised §4 confirming that δ(k,6)≥δ_6>0 with δ_6 independent of k, thereby justifying the o(e(widetilde{O}_{k+1})) claim as k→∞. revision: yes

  2. Referee: [§5, Corollary 5.1] the claimed Ramsey-type implication is stated only as a one-sentence consequence; it is unclear whether the o() density forces the Ramsey number to be finite or merely sublinear, and no quantitative relation between the two statements is supplied.

    Authors: We agree that the one-sentence remark is too terse. The o() density bound directly yields that any subgraph of widetilde{O}_{k+1} with positive edge density (in the limit k→∞) must contain a C_{2r} for r≥6; this translates into a Ramsey-type statement that the largest C_{2r}-free subgraph in the family has size o(total edges). We will expand Corollary 5.1 with a precise formulation, clarifying that the consequence is an o(N) upper bound on the size of the largest C_{2r}-free subgraph (rather than a finite Ramsey number in the classical sense) and supplying the explicit quantitative link from the coefficient in Theorem 4.3. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states explicit upper bounds on ex(J(n;k,k+1),C_{2r}) derived from the bipartition and adjacency structure of the doubled Johnson graphs together with standard cycle-counting and deletion methods. No equation, parameter fit, or central claim reduces by construction to its own inputs; the o(e(tilde O_{k+1})) statement for r≥6 is presented as a consequence of the derived bound rather than a tautological restatement. No self-citation is invoked as a load-bearing uniqueness theorem or ansatz source. The derivation chain is therefore self-contained against external combinatorial techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, ad-hoc axioms, or invented entities are visible.

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Reference graph

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