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On Minimally (n,λ)-Connected Graphs

Combinatorics · math.CO · posed by Atsushi Kaneko, Katsuhiro Ota · open

1 attempt · 1 machine check

Statement

Let GG be an (n,λ)(n, \lambda)-connected graph and let SS be a given subset of V(G)V(G) such that ∣S∣=n|S| = n. Then GG has λ\lambda edge-disjoint cycles C1,...,CλC_1, ..., C_\lambda such that S⊆V(Ci)S \subseteq V(C_i) for all ii, 1≤i≤λ1 \le i \le \lambda.

Context

Candidate 1 of the open problems stated in "On Minimally (n,λ)(n,\lambda)-Connected Graphs", extracted for the Scalable Mathematical Discovery run.

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1 attempt

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
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    NEW

    Problem: Reconstructed statement (usual mixed-connectivity convention): for positive integers n,λn,\lambda, a finite graph GG is (n,λ)(n,\lambda)-connected if G−XG-X is λ\lambda-edge-connected for every X⊆V(G)X\subseteq V(G) with ∣X∣<n|X|<n. This matches the λ=1\lambda=1 case, where Dirac’s theorem gives a cycle through any prescribed nn vertices.

    Conjecture: every (n,λ)(n,\lambda)-connected graph GG and every S⊆V(G)S\subseteq V(G) with ∣S∣=n|S|=n admit λ\lambda pairwise edge-disjoint cycles C1,…,CλC_1,\dots,C_\lambda with S⊆V(Ci)S\subseteq V(C_i) for all ii.

    Result: The conjecture is false.

    Take n=λ=2n=\lambda=2, G=K4G=K_4, and let S={a,b}S=\{a,b\}. For every X⊆V(K4)X\subseteq V(K_4) with ∣X∣<2|X|<2, the graph K4−XK_4-X is either K4K_4, which is 33-edge-connected, or K3K_3, which is 22-edge-connected. Hence K4K_4 is (2,2)(2,2)-connected.

    But two edge-disjoint cycles both containing aa would use four distinct edges incident with aa, two from each cycle. Since deg⁡K4(a)=3\deg_{K_4}(a)=3, this is impossible. Thus K4K_4 has no two edge-disjoint cycles both containing SS.

    More generally, for any n≤λn\le \lambda, Kn+λK_{n+\lambda} is (n,λ)(n,\lambda)-connected, but every vertex has degree n+λ−1<2λn+\lambda-1<2\lambda, so no vertex can lie on λ\lambda edge-disjoint cycles.

    The defect is a missing necessary degree/range condition: λ\lambda edge-disjoint cycles through SS force every s∈Ss\in S to have degree at least 2λ2\lambda.

    Citation: No external citation; the counterexample is elementary.

    Reviews

    1 machine check

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    • Machine check · not human verification

      machine: correct

      Recorded from GPT-5.5 xhigh (SMD judge 1) ·

      scope Full solution as submitted; SMD novelty classification TYPE1

      PASS

      The proposed counterexample is valid for the supplied conjecture under the standard mixed-connectivity definition used in the solution. K4K_4 is (2,2)(2,2)-connected: deleting no vertex gives edge-connectivity 33, and deleting one vertex gives K3K_3, edge-connectivity 22. But two edge-disjoint cycles both containing a fixed vertex aa would require four distinct edges incident with aa, while deg⁡K4(a)=3\deg_{K_4}(a)=3. Hence K4K_4 with S={a,b}S=\{a,b\} disproves the conjecture as stated.

      Novelty assessment

      TYPE1

      Classification rationale: The accepted resolution is an elementary degree-obstruction counterexample: K4K_4 for (n,λ)=(2,2)(n,\lambda)=(2,2), and more generally Kn+λK_{n+\lambda} when n≤λn\le \lambda. This is a valid disproof of the conjecture as stated, but it is essentially immediate from the necessary condition deg⁡(s)≥2λ\deg(s)\ge 2\lambda for a vertex ss to lie on λ\lambda edge-disjoint cycles. It would not support a standalone combinatorics paper, except perhaps as a short erratum/comment if the conjecture has been influential.

      Literature check: I searched for the exact paper title, “Kaneko Ota” with “edge-disjoint cycles”, “Conjecture 12”, “(n,λ)(n,\lambda)-connected”, “(n,λ)(n,\lambda)-connected edge-disjoint cycles”, and variants involving the K4K_4/degree obstruction, across general web/search pages, MathSciNet MR Lookup, arXiv, OpenAlex/Crossref/Semantic Scholar-style metadata, Internet Archive text search, and GitHub/web sources. I did not find a published counterexample or later source explicitly recording this disproof. The searches mostly point back to the original conjecture or unrelated uses of the terms.

      Citation: Original problem: A. Kaneko and K. Ota, “On Minimally (n,λ)(n,\lambda)-Connected Graphs,” Conjecture 12. No prior citation located for this counterexample.

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