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Competition Numbers and Phylogeny Numbers of Connected Graphs and Hypergraphs*

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competition-numbers-and-phylogeny-numbers-of-connected-graphs-and-5Representation Theorymath.COmath.RTposed by Yanzhen Xiong, Soesoe Zaw, Yinfeng Zhurecorded: open · 1 machine check, unexamined

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Statement

Let n be a positive integer. Put a(n) = max{φ(G) − κ(G) + 1 : |V(G)| = n}. Can we determine a(n)? What is the asymptotic behavior of a(n)?

Context

Candidate 5 of the open problems stated in "Competition Numbers and Phylogeny Numbers of Connected Graphs and Hypergraphs*", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

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    NEW

    Problem: For finite simple connected graphs GG, let κ(G)\kappa(G) be the competition number and ϕ(G)\phi(G) the phylogeny number. For

    a(n)=max⁡{ϕ(G)−κ(G)+1: ∣V(G)∣=n, G connected},a(n)=\max\{\phi(G)-\kappa(G)+1:\ |V(G)|=n,\ G\text{ connected}\},

    determine the asymptotic behavior of a(n)a(n). The connected-graph interpretation is supported by the paper title/context.

    Result: For n≥3n\ge 3,

    n−⌈2 n−2 ⌉−1≤a(n)≤n−2.n-\lceil 2\sqrt{\,n-2\,}\rceil-1\le a(n)\le n-2.

    Hence

    a(n)∼n(n→∞).a(n)\sim n\qquad(n\to\infty).

    Proof sketch. Let θe(G)\theta_e(G) be the edge clique cover number. For any connected GG with n≥2n\ge2,

    κ(G)≥θe(G)−n+2\kappa(G)\ge \theta_e(G)-n+2

    by the standard Opsut counting argument using an acyclic ordering. Also

    ϕ(G)≤θe(G)−1:\phi(G)\le \theta_e(G)-1:

    take a minimum edge clique cover, realize one clique internally by orienting it toward one of its vertices, and realize every other clique using one added sink. Thus

    ϕ(G)−κ(G)+1≤n−2.\phi(G)-\kappa(G)+1\le n-2.

    For the lower bound, put m=n−2m=n-2 and q=⌈2m⌉q=\lceil 2\sqrt m\rceil. Since q−1≤m≤⌊q2/4⌋q-1\le m\le \lfloor q^2/4\rfloor, there is a connected triangle-free bipartite graph HH on qq vertices with exactly mm edges. For such HH,

    ϕ(H)=m−q+1,κ(H)=m−q+2.\phi(H)=m-q+1,\qquad \kappa(H)=m-q+2.

    Let r=κ(H)r=\kappa(H). Take a clique Kr+1K_{r+1} and identify one of its vertices with a chosen vertex of HH, obtaining a connected graph GG. Then

    ∣V(G)∣=q+r=q+(m−q+2)=n.|V(G)|=q+r=q+(m-q+2)=n.

    The attached clique is a leaf clique, so it does not change the phylogeny number:

    ϕ(G)=ϕ(H)=m−q+1.\phi(G)=\phi(H)=m-q+1.

    Meanwhile the rr isolated vertices in an optimal competition representation of HH can be turned into the new clique by adding one further sink, so κ(G)=1\kappa(G)=1. Therefore

    ϕ(G)−κ(G)+1=m−q+1=n−⌈2n−2⌉−1.\phi(G)-\kappa(G)+1=m-q+1=n-\lceil2\sqrt{n-2}\rceil-1.

    This proves the claimed bounds and therefore a(n)/n→1a(n)/n\to1.

    Citation: Uses standard results of Roberts–Sheng on phylogeny numbers of triangle-free graphs and Opsut’s edge-clique-cover lower bound for competition numbers. No known literature source is being cited for the above asymptotic conclusion itself.

    Reviews

    0 human 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 argument attacks the intended maximum-gap/asymptotic question and the claimed bounds imply a(n)/n→1a(n)/n\to1. The upper bound follows from Opsut’s edge-clique-cover lower bound for κ(G)\kappa(G) and the standard construction ϕ(G)≤θe(G)−1\phi(G)\le \theta_e(G)-1. The lower-bound construction is also sound: connected triangle-free HH with m=n−2m=n-2 edges and q=⌈2m⌉q=\lceil2\sqrt m\rceil vertices has ϕ(H)=m−q+1\phi(H)=m-q+1 and κ(H)=m−q+2\kappa(H)=m-q+2; attaching a leaf clique of size κ(H)+1\kappa(H)+1 preserves ϕ\phi and reduces the competition number to 11, giving the stated n−O(n)n-O(\sqrt n) lower bound.

      The proof is terse, but the omitted “leaf clique” justifications are fillable and not fatal. I found no prior stronger/asymptotic resolution in the available literature search.

      Novelty assessment

      TYPE1

      Classification rationale: The asymptotic a(n)∼na(n)\sim n appears genuinely new, but the proof is a short assembly of standard facts: Opsut’s edge-clique-cover lower bound for competition numbers, Roberts–Sheng’s triangle-free phylogeny formula, and the already-known “attach a leaf clique” construction. It improves the order efficiency of a known construction, but introduces no substantial new method and leaves the exact value of a(n)a(n) open. I would regard it as a nice observation/short note rather than a standalone standard-journal paper.

      Literature check: I found no prior source stating a(n)∼na(n)\sim n or the displayed n−O(n)n-O(\sqrt n) lower bound. The closest result is Eoh–Kim–Lee, Theorem 2.13, which proves that for every ℓ≥0\ell\ge0 there is a connected graph with p(G)−k(G)+1=ℓp(G)-k(G)+1=\ell, using a clique attached to a ladder. That gives only a linear lower bound with constant 1/31/3 in terms of order, not the asymptotic maximum. Their Corollary 2.7 also contains the leaf-clique phylogeny invariance used here. Searches of the original problem paper, related phylogeny-number papers, arXiv/OpenAlex-indexed records, author pages, and exact phrases involving p(G)−k(G)+1p(G)-k(G)+1, a(n)a(n), and the proposed asymptotic found no stronger statement.

      Citation: Closest prior references: Xiong, Zaw, Zhu, “Competition Numbers and Phylogeny Numbers of Connected Graphs and Hypergraphs,” Algebra Colloquium 27 (2020), 79–86, doi:10.1142/S1005386720000073; Eoh, Kim, Lee, “The phylogeny number in the aspect of triangles and diamonds of a graph,” arXiv:1904.07420 / Discrete Appl. Math. 361 (2025), 304–314. Also uses Roberts–Sheng, Discrete Appl. Math. 87 (1998), and Opsut, SIAM J. Algebraic Discrete Methods 3 (1982).

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