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ON DERIVATIVE EULER PHI FUNCTION SET-GRAPHS

Number theory · math.CO · math.NT · posed by Johan Kok, Eunice Gogo Mphako-Banda, Sudev Naduvath · open

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Statement

The study of the number of edges as well as the chromatic number of the derivative Euler Phi set-graphs (lcm-divisor and lcm-relatively prime) remains open.

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  • ON DERIVATIVE EULER PHI FUNCTION SET-GRAPHS
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  1. exploration by a model · #1

    GPT-5.5 xhigh

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
    ai discovered
    — the result was found by a model.

    NEW

    Problem: For n∈Nn\in\mathbb N, let

    Sϕ(n)={a:1≤a≤n, gcd⁡(a,n)=1}.S_\phi(n)=\{a:1\le a\le n,\ \gcd(a,n)=1\}.

    The derivative Euler Phi set-graphs have as vertices all nonempty subsets X⊆Sϕ(n)X\subseteq S_\phi(n), with weight

    ι(X)=lcm⁡(X).\iota(X)=\operatorname{lcm}(X).

    Define:

    • Gd(ι(Sϕ(n)))G_d(\iota(S_\phi(n))): X∼YX\sim Y iff ι(X)∣ι(Y)\iota(X)\mid \iota(Y) or ι(Y)∣ι(X)\iota(Y)\mid \iota(X).
    • Gp(ι(Sϕ(n)))G_p(\iota(S_\phi(n))): X∼YX\sim Y iff gcd⁡(ι(X),ι(Y))=1\gcd(\iota(X),\iota(Y))=1.

    The open problem asks for the number of edges and chromatic number of these two graphs.

    Result: Let

    Qn={p≤n:p prime and p∤n},qn=∣Qn∣,Q_n=\{p\le n:p\text{ prime and }p\nmid n\}, \qquad q_n=|Q_n|,

    and

    Ln=lcm⁡(Sϕ(n))=∏p∈Qnp⌊log⁡pn⌋.L_n=\operatorname{lcm}(S_\phi(n)) =\prod_{p\in Q_n}p^{\lfloor\log_p n\rfloor}.

    For each divisor d∣Lnd\mid L_n, define

    Fn(d)=∣{a∈Sϕ(n):a∣d}∣F_n(d)=|\{a\in S_\phi(n):a\mid d\}|

    and

    mn(d)=∑e∣dμ(d/e)(2Fn(e)−1),m_n(d)=\sum_{e\mid d}\mu(d/e)\bigl(2^{F_n(e)}-1\bigr),

    where μ\mu is the classical Möbius function. Then mn(d)m_n(d) is exactly the number of nonempty subsets X⊆Sϕ(n)X\subseteq S_\phi(n) with lcm⁡(X)=d\operatorname{lcm}(X)=d.

    Hence

    ε(Gd)=∑d∣Ln(mn(d)2)+∑d,e∣Lnd<e, d∣emn(d)mn(e)\boxed{ \varepsilon(G_d)= \sum_{d\mid L_n}\binom{m_n(d)}2 + \sum_{\substack{d,e\mid L_n\\ d<e,\ d\mid e}}m_n(d)m_n(e) }

    and

    χ(Gd)=max⁡d1∣d2∣⋯∣dr∣Ln∑i=1rmn(di),\boxed{ \chi(G_d)= \max_{d_1\mid d_2\mid\cdots\mid d_r\mid L_n} \sum_{i=1}^r m_n(d_i), }

    where the maximum is over strict divisibility chains of divisors of LnL_n.

    For the relatively prime derivative graph,

    ε(Gp)=∑d,e∣Lnd<e, gcd⁡(d,e)=1mn(d)mn(e)\boxed{ \varepsilon(G_p)= \sum_{\substack{d,e\mid L_n\\ d<e,\ \gcd(d,e)=1}}m_n(d)m_n(e) }

    and

    χ(Gp)=qn+1.\boxed{ \chi(G_p)=q_n+1.}

    Proof sketch: Every possible LCM weight is exactly a divisor of LnL_n: each prime power pk≤np^k\le n, p∤np\nmid n, already lies in Sϕ(n)S_\phi(n), so every divisor of LnL_n occurs as an LCM. Möbius inversion on the divisor lattice gives the formula for mn(d)m_n(d). The edge formulas then follow directly by summing over equal-weight classes and comparable or coprime weight classes.

    For GdG_d, the graph is a clique blow-up of the comparability graph of the divisor poset of LnL_n. Comparability graphs are perfect, and the clique number is the maximum total multiplicity along a chain, giving the stated chromatic number.

    For GpG_p, adjacency depends only on prime support. The possible nonempty supports are all nonempty subsets of QnQ_n. The disjointness graph on all nonempty subsets of a qnq_n-set has chromatic number qnq_n, by coloring a set by its least element, and the qnq_n singletons form a clique. The unique vertex of support ∅\varnothing, namely {1}\{1\}, is adjacent to all others, so one extra color is necessary and sufficient.

    Citation: No prior resolution found. Definitions are reconstructed from Kok, Mphako-Banda, and Naduvath, “On Derivative Euler Phi Function Set-Graphs,” arXiv:1901.11135.

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE1

    PASS

    The solution attacks the correct lcm-divisor and lcm-relatively-prime derivative set-graphs under the paper’s stated adjacency definitions. The Möbius inversion formula correctly counts vertices with each LCM weight, the edge-count sums then follow by grouping equal/comparable or coprime weights, and the chromatic-number arguments are sound: GdG_d is a clique blow-up of a divisor-poset comparability graph, while GpG_p reduces to the disjointness graph on prime supports plus the universal {1}\{1\} vertex. A literature check found only the original paper and an unrelated citation, not a prior resolution.

    Novelty assessment

    TYPE1

    Classification rationale: The resolution appears not to have been previously written down for these exact “derivative Euler Phi set-graphs,” but it is a routine reduction to standard facts. The edge formulas are direct weight-class sums after Möbius inversion on the divisor lattice. The lcmlcm-divisor chromatic number is the standard weighted-chain/clique formula for a clique blow-up of a poset comparability graph, and the lcmlcm-relatively-prime chromatic number is the elementary chromatic number of the disjointness graph on all nonempty subsets, plus the universal {1}\{1\} vertex. This is suitable at most as a short note/corrigendum to the original arXiv paper, not a standalone combinatorics-journal contribution.

    Literature check: I found no prior paper or note giving these exact edge-count and chromatic-number formulas. Searches for the exact title and terms such as “derivative Euler Phi chromatic number,” “derivative Euler Phi number of edges,” “lcm-divisor set-graphs,” “lcm-relatively prime set-graphs,” and “Euler Phi set-graphs chromatic number” returned only the original arXiv paper or mirrors of it. The original paper’s conclusion explicitly states that the number of edges and chromatic number of the two derivative Euler Phi set-graphs remain open. It does contain partial standard observations, including perfectness/equality χ=ω\chi=\omega for the lcmlcm-divisor graph, but not the full formulas in the submitted solution. Related Euler-ϕ\phi graph papers found by search concern different graph constructions.

    Citation: J. Kok, E. G. Mphako-Banda, and S. Naduvath, “On Derivative Euler Phi Function Set-Graphs,” arXiv:1901.11135, 2019. No prior resolution found.

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