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Graphs with characteristic-dependent well-covered dimension

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graphs-with-characteristic-dependent-well-covered-dimensionNumber Theorymath.NTmath.RTposed by Joseph Burdickrecorded: open · 1 machine check, unexamined

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Statement

Based upon the results generated from our Sage script, we submit as a conjecture that these graphs constructed be the smallest graphs (by order) that have characteristic-dependent well-covered dimension for any given characteristic.

Context

Candidate 1 of the open problems stated in "Graphs with characteristic-dependent well-covered dimension", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
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    NEW

    Problem: For a finite simple graph GG and field FF, let wcdim⁡(G,F)\operatorname{wcdim}(G,F) be the dimension over FF of the vector space of vertex weightings for which all maximal independent sets have the same total weight. The conjecture is naturally formalized as: for each prime pp, the smallest order of a graph whose well-covered dimension changes in characteristic pp is p+5p+5, the order supplied by Burdick’s construction.

    Result: The conjecture is false. We give a characteristic-1313 counterexample on 16<18=13+516<18=13+5 vertices.

    For positive integers m,nm,n, define Hm,nH_{m,n} with vertices

    Y={y1,…,yn},U={u1,…,un},V={v1,…,vm},W={w1,…,wm}.Y=\{y_1,\dots,y_n\},\quad U=\{u_1,\dots,u_n\},\quad V=\{v_1,\dots,v_m\},\quad W=\{w_1,\dots,w_m\}.

    Edges are:

    1. Y∪VY\cup V is a clique;
    2. every uiu_i is adjacent to every waw_a;
    3. vav_a is adjacent to waw_a;
    4. yiy_i is adjacent to uju_j for i≠ji\ne j.

    The maximal independent sets are exactly

    Mi={yi}∪W,Ni={yi,ui},M_i=\{y_i\}\cup W,\qquad N_i=\{y_i,u_i\}, La={va}∪(W∖{wa}),Ka={va}∪U.L_a=\{v_a\}\cup (W\setminus\{w_a\}),\qquad K_a=\{v_a\}\cup U.

    Indeed, an independent set contains at most one vertex from the clique Y∪VY\cup V. If it contains yiy_i, maximality forces either uiu_i or all of WW. If it contains vav_a, maximality forces either all of UU or all of W∖{wa}W\setminus\{w_a\}. If it contains no vertex of Y∪VY\cup V, it is not maximal.

    Let a weighting have values

    αi=w(yi),βi=w(ui),γa=w(va),δa=w(wa),\alpha_i=w(y_i),\quad \beta_i=w(u_i),\quad \gamma_a=w(v_a),\quad \delta_a=w(w_a),

    and put B=∑iβiB=\sum_i\beta_i, D=∑aδaD=\sum_a\delta_a. If all maximal independent-set sums equal λ\lambda, then

    αi+D=αi+βi=λ,\alpha_i+D=\alpha_i+\beta_i=\lambda,

    so βi=D\beta_i=D for all ii, hence B=nDB=nD. Also

    γa+D−δa=γa+B=λ,\gamma_a+D-\delta_a=\gamma_a+B=\lambda,

    so δa=D−B=(1−n)D\delta_a=D-B=(1-n)D. Summing over aa,

    D=m(1−n)D,D=m(1-n)D,

    equivalently

    (m(n−1)+1)D=0.\bigl(m(n-1)+1\bigr)D=0.

    Conversely, these equations define all well-covered weightings. Therefore

    wcdim⁡(Hm,n,F)={1,char⁡(F)∤m(n−1)+1,2,char⁡(F)∣m(n−1)+1.\operatorname{wcdim}(H_{m,n},F)= \begin{cases} 1,& \operatorname{char}(F)\nmid m(n-1)+1,\\ 2,& \operatorname{char}(F)\mid m(n-1)+1. \end{cases}

    Now take m=3,n=5m=3,n=5. Then

    m(n−1)+1=3⋅4+1=13m(n-1)+1=3\cdot4+1=13

    and

    ∣V(H3,5)∣=2m+2n=6+10=16.|V(H_{3,5})|=2m+2n=6+10=16.

    Thus

    wcdim⁡(H3,5,Q)=1,wcdim⁡(H3,5,F)=2\operatorname{wcdim}(H_{3,5},\mathbb Q)=1, \qquad \operatorname{wcdim}(H_{3,5},F)=2

    for every field FF of characteristic 1313. Hence characteristic-dependent well-covered dimension occurs for characteristic 1313 on 1616 vertices, contradicting the conjectured minimum 13+5=1813+5=18.

    Citation: Conjecture source: Joseph Burdick, “Graphs with characteristic-dependent well-covered dimension,” Rose-Hulman Undergraduate Mathematics Journal 17(1), Article 10, 2016; arXiv version with Oscar Vega, arXiv:1506.00180. The counterexample above is self-contained.

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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 construction attacks the correct minimality conjecture: Burdick–Vega’s Theorem 2 gives order p+5p+5 for odd characteristic pp, and the conjecture speculates these are smallest.

      The graph H3,5H_{3,5} is rigorously analyzed. The listed maximal independent sets are exhaustive, and the well-covered weighting equations correctly give

      (m(n−1)+1)D=0.(m(n-1)+1)D=0.

      For m=3,n=5m=3,n=5, this is 13D=013D=0, so the well-covered dimension is 11 in characteristic ≠13\ne 13 and 22 in characteristic 1313. The graph has 2m+2n=16<18=13+52m+2n=16<18=13+5 vertices, so it is a valid counterexample.

      I found no prior similar or stronger resolution in the relevant literature searches; only the original Burdick–Vega paper appeared.

      Novelty assessment

      TYPE1

      Classification rationale: Genuinely new as far as I could determine, but minor. The construction is a short, elementary two-parameter variation of Burdick–Vega’s original family, giving a 16-vertex counterexample for characteristic 13. It disproves an obscure minimality conjecture from an undergraduate journal, but is likely too small/routine for a standalone standard combinatorics journal paper.

      Literature check: I found only the original Burdick/RHUMJ paper and mirrors/indexing pages (arXiv, Semantic Scholar, ADS, ResearchGate/Scispace). Searches for “characteristic-dependent well-covered dimension,” “well-covered dimension characteristic,” “characteristic 13 well-covered dimension,” “m(n-1)+1 well-covered,” and related Burdick/Vega/minimality terms did not reveal the counterexample or a stronger known construction. The only other arXiv hit centered on “well-covered dimension” was Clemente’s unrelated work on minimal well-covered dimension.

      Citation: Joseph Burdick and Oscar Vega, “A note on graphs with characteristic-dependent well-covered dimension,” arXiv:1506.00180; Joseph Burdick, “Graphs with characteristic-dependent well-covered dimension,” Rose-Hulman Undergraduate Mathematics Journal 17(1), Article 10, 2016.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

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