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On Sierpi\'{n}ski packing chromatic number and recognition of Sierpi\'{n}ski products

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper determines the Sierpiński packing chromatic number exactly for products of complete graphs, shows path and star products always pack with three colors, and gives a polynomial-time recognition test for tree products.

desk verdict Novel setup and a solid recognition algorithm, but the path upper-bound proof has a load-bearing error and Theorem 3.5 has a gap; the paper needs major revision, not desk rejection. read the letter →

arxiv 2507.06730 v1 pith:SSGON2DJ submitted 2025-07-09 math.CO

Přemysl Holub , Sandi Klavžar This is my paper · ORCID
classification math.CO MSC 05C7605C1505C1268Q25
keywords Sierpińskiproductpackingchromaticnumberuppercompletegraphspathsandstarstreerecognitioncoronacoloring
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper defines the Sierpiński packing chromatic number $\chi_\rho(G,H)$ as the minimum, over all linking functions $f\colon V(G)\to V(H)$, of the packing chromatic number of the Sierpiński product $G\otimes_f H$, and the upper Sierpiński packing chromatic number $b\chi_\rho(G,H)$ as the corresponding maximum. Its central result is exact: for $m,n\ge 3$, $\chi_\rho(K_m,K_n)=mn-2m+2$, while $b\chi_\rho(K_m,K_n)$ equals $mn-2m+2$ when $n\ge m$ and $mn-m-n+2$ when $m\ge n$. The paper also proves that Sierpiński products of two paths or of a path and a star always have packing chromatic number $3$, bounds the worst case for those products by small constants ($6$ or $7$ for two paths under mild length conditions), and gives a polynomial-time algorithm to recognize graphs that are Sierpiński products of two trees. A careful reader cares because computing the packing chromatic number is hard even for trees, so exact best- and worst-case values over an entire product family are rare and concrete.

What carries the argument

The central object is the Sierpiński product $G\otimes_f H$: its vertex set is $V(G)\times V(H)$, it contains a disjoint copy of $H$ over each vertex of $G$ (type-1 edges), and for each edge $gg'$ of $G$ it has one connecting edge $(g,f(g'))(g',f(g))$ (type-2 edges). The argument's engine is a packing coloring, in which every pair of vertices sharing color $i$ must be at distance greater than $i$, so each color class is an $i$-packing. For complete factors the proof uses three structural facts: the product has diameter $3$; every fiber is a clique, so no color can appear more than $m$ times; and the extremal colorings are built from one independent set and one $2$-packing of size $m$, with all other vertices left to private colors. For path and star factors the lower bounds are obtained by embedding corona graphs $P_n\odot 2K_1$ and $P_n\odot 3K_1$ as induced subgraphs, where the needed packing chromatic numbers are already known. The recognition algorithm rests on Lemma 5.1, which characterizes a pendant connecting edge as exactly a cut edge whose removal leaves components of orders $n(T)$ and $n(T\otimes_f H)-n(T)$; this lets the algorithm peel fiber copies and test tree isomorphism at each step.

What would settle it

Take the tree consisting of a two-vertex spine with one pendant path of length 2 attached to one spine vertex and apply the Lemma 4.2 pattern: the spine vertex receives color 1 and the first path vertex also receives color 1, so two adjacent vertices lie in the same color class, which is not a packing coloring. Checking this single small tree falsifies the lemma as stated and thereby removes the current proof of the upper bound $b\chi_\rho(P_m,P_n)\le 7$.

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Extended reading notes

Core claim

For a fixed pair $G,H$, every choice of $f$ gives a different graph $G\otimes_f H$, so the paper studies the range of the packing chromatic number across that family. The main discovery is that for complete factors the range is fully determined by the two parameters $m=|V(K_m)|$ and $n=|V(K_n)|$: the lower bound $mn-2m+2$ comes from the fact that each fiber is a clique (so at most $m$ vertices can share any color), the whole product has diameter $3$ (so colors larger than $2$ cannot be reused at all), and the bound is attained by a coloring that reserves one color class of size $m$ and one 2-packing of size $m$, then gives every remaining vertex a private color. The upper Sierpiński value is the same when the fiber is at least as large as the base, and is $mn-m-n+2$ when the base is larger, attained by a function that maps $n$ base vertices onto distinct fiber vertices and the remaining base vertices onto one fixed vertex. For paths and stars the paper shows that the minimum is always $3$ across all four factor combinations, while the maximum is bounded by $7$ for path–path and star–path products and by $9$ for path–star products, with sharper interval statements in several ranges. The recognition result is stated as Theorem 5.2: a connected graph can be tested in polynomial time for being isomorphic to $T_1\otimes_f T_2$ for trees $T_1,T_2$, by using pendant connecting cut edges to peel off fiber copies one at a time.

Load-bearing premise

The load-bearing premises are that the Lemma 4.2 coloring pattern on the tree class is a legal packing coloring (the pattern assigns the first vertex of each attached path the same color as the adjacent spine vertex, placing two distance-1 vertices in one color class), and that in the $m\ge n$ case of Theorem 3.5 the product $K_m\otimes_f K_n$ contains disjoint independent and 2-packing sets of size $m$ for non-surjective $f$; the proof of the latter does not supply the required construction.

Editorial extensions

If this is right

  • For $m,n\ge 3$, the Sierpiński packing chromatic number of complete pairs is exactly $mn-2m+2$; every linking function uses at least that many colors and one linking function achieves it.
  • For $m\ge 12$, $n\ge 3$, every Sierpiński product of $P_m$ and $P_n$ can be packed with at most 7 colors and at least one product needs 6, so the upper Sierpiński value is 6 or 7.
  • For the four path/star combinations, the Sierpiński packing chromatic number is exactly 3, so the best linking function never needs more than three colors; the worst case stays within 7 or 9.
  • The recognition theorem gives a polynomial-time test for whether a given connected graph is a Sierpiński product of two trees, using the cut-edge structure of pendant connecting edges.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: if the same two invariants—independence number and 2-packing number—are determined for other factor pairs, the complete-graph proof pattern suggests closed formulas for their Sierpiński packing chromatic numbers may follow directly.
  • Beyond the paper: the peeling-by-cut-edges recognition strategy suggests a testable extension to factors that are not trees, for example any base graph with a known pendant-edge decomposition, where component orders after cut-edge removal would still identify fibers.
  • Beyond the paper: the three-colour results for stars and paths indicate that sparse factor pairs with small diameter-2 neighborhoods may always admit a three-colour linking function; testing complete bipartite factors would be a natural next step.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper introduces the Sierpiński packing chromatic number chi_rho(G,H) and the upper Sierpiński packing chromatic number bchi_rho(G,H), defined as the minimum and maximum of chi_rho(G ⊗_f H) over all connecting functions f. It claims exact values for complete graph factors: chi_rho(K_m,K_n)=mn-2m+2 for m,n≥3, and bchi_rho(K_m,K_n)=mn-2m+2 for n≥m and mn-m-n+2 for m≥n. It further claims that all four Sierpiński products of paths and stars have Sierpiński packing chromatic number 3, with upper bounds on the upper Sierpiński packing chromatic number, in particular bchi_rho(P_m,P_n)∈{6,7} for n≥3, m≥12. Finally, it gives a polynomial-time recognition algorithm for graphs that are Sierpiński products of two trees.

Significance. If the results were established, the complete graph formulas and the path/star bounds would be a useful contribution to the packing coloring literature, and the recognition algorithm would be a novel algorithmic contribution. The paper has several strengths: the definitions are clear, the lower bounds via corona theorems are sound in isolation, and the recognition algorithm in Section 5 is plausible and appears to be the most robust part of the manuscript. However, the submitted proofs contain load-bearing gaps, and at least one stated value in the complete graph section is false. The central claims for m,n≥3 may be salvageable, but the present derivations do not support them.

major comments (4)
  1. [Lemma 4.2 and Theorem 4.4] The coloring defined in Lemma 4.2 is not a packing coloring. For each attached path Q_i, the first vertex of Q_i receives the same color c(v_i) as the spine vertex v_i to which it is adjacent. Thus two vertices of the same color are at distance 1, which violates the packing condition d>ell for every color ell≥1. Since Lemma 4.3 shows that P_m ⊗_f P_n belongs to the class T, the upper bound bchi_rho(P_m,P_n)≤7 in Theorem 4.4 relies entirely on this defective lemma. The lower bound bchi_rho(P_m,P_n)≥6 via P_m ⊙ 2K_1 appears sound, but the claim {6,7} is not established as written.
  2. [Theorem 3.5, non-surjective f] In the non-surjective case of Theorem 3.5, the proof asserts, by reference to Case 1 of Lemma 3.3, that G contains a disjoint 2-packing and an independent set both of size m. This inference is invalid. The referenced argument invokes Lemma 3.2, which requires the order of the second factor to be at least the order of the first; after deleting the unused fiber vertex, the second factor has order n-1 < m when m≥n. The claim is in fact false for K_4 ⊗_f K_3 with f=(1,1,2,2): the only 2-packing of size 4 is {(u_i,3): i∈[4]}, but there is no independent set of size 4 in the remaining graph disjoint from it. A direct check shows that selecting (u_1,1) forces (u_2,2) and then both (u_3,2) and (u_4,2), which are adjacent, while selecting (u_1,2) forces (u_3,2) and (u_4,2), which are adjacent. This case is needed for the upper bound in Theorem 3.5, so the theorem is not proved as written.
  3. [Lemma 3.3, Case 1] Lemma 3.3 also misapplies Lemma 3.2 when n=m and f is non-surjective, since after removing the unused vertex the second factor has order n-1 < m. Moreover, the proof as written does not assign a color to the set M={(u_i,n)} and then colors all remaining uncolored vertices with distinct colors, which would use more than mn-2m+2 colors. The intended argument presumably colors M with color 2, but even with that repair, the existence of an independent set of size m in G' requires a new proof that is not supplied.
  4. [Section 3, case m=2] The paragraph after Theorem 3.5 states that for m=2 the graph K_m ⊗_f K_n is 'a diameter 2 graph with α(G)=2, hence χ_ρ(G)=2n−1'. This is false for n≥3. The graph is two disjoint K_n cliques joined by a single edge and has diameter 3. For n=3, χ_ρ(K_2,K_3)=4: color one non-connecting vertex from each clique with 1, the other two non-connecting vertices with 2, and the two connecting vertices with 3 and 4. A 3-coloring is impossible because colors 1 and 2 can be used on at most two vertices each and color 3 can be used at most once, covering at most 5 of the 6 vertices. Consequently the claimed determination of the complete graph case for all m,n is not correct as stated. The same subsection's Proposition 3.6 also gives bchi_rho(K_3,K_2)=3, whereas the preceding case analysis (1) yields 4 for m=3.
minor comments (4)
  1. [Lemma 3.1] In the constant-function case of Lemma 3.1, the proof chooses vertices (u_j,i') and (u_l,i'') with i'≠i''; when n=2 no two distinct values different from the constant value exist. Choosing i'=i'' (the unique value different from the constant value) still gives distance 3, so the proof can be repaired.
  2. [Theorem 4.1] In the proof of Theorem 4.1, the notation V(K_m) appears where V(P_m) is intended; the base graph in that theorem is a path.
  3. [Lemma 5.1] The expression T ⊗_f |_{T-x} H is not defined; it should be something like (T-x) ⊗_{f|_{V(T-x)}} H.
  4. [Theorem 4.6] The 64-term coloring pattern on the shortest path Q is said to be 'found and verified by a computer', but no certificate or reproducible verification is provided; the proof should include a human-checkable certificate or the code used.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's derivations rest on the definition of Sierpiński products, packing-color axioms, and cited external theorems; the flagged Lemma 4.2 defect is a proof error, not a circular reduction.

full rationale

The paper's derivation chain is self-contained in the relevant sense. The main results for complete factors (Theorems 3.4 and 3.5) are proven from Lemma 3.1 (diameter 3), Lemma 3.2 (α2 = m), and Lemma 3.3's counting argument: every fiber is a clique, so at most m vertices may receive color 1 and at most m color 2, and every other vertex needs a private color. This is a direct lower-bound argument from the definitions, not a fitted quantity renamed as a prediction. The upper bounds are constructive colorings exhibited in the paper (e.g., the independent set X and 2-packing X′ in Theorem 3.4, and the explicit f′ and packing-size analysis in Theorem 3.5). No parameter is fitted to data and then 'predicted'; no conclusion is assumed in its own premises; no load-bearing claim is justified only by a self-citation. The path and star sections use cited external theorems on coronae (Theorems 2.1–2.3, from Laïche–Bouchemakh–Sopena) and construct explicit colorings; the upper bound bχρ(Pm,Pn) ≤ 7 rests on Lemma 4.2, whose proposed coloring incident to attached paths is indeed flawed (it gives the first attached vertex the same color as its adjacent spine vertex, violating the packing condition). That is a mathematical correctness gap in a proof, not a circularity: Lemma 4.2 does not assume the conclusion it is used to prove, and Theorem 4.4's lower bound is independently supplied by an induced corona with known chromatic number. Likewise, the Theorem 3.5 non-surjective case appears to import Lemma 3.2 beyond its stated hypothesis, which is a technical gap, not a circular definition or fitted prediction. The polynomial-time recognition theorem is an algorithmic reduction (checking cut-edge components and tree isomorphism), not a circular argument. Overall, I find no instance in which a claimed result reduces, by the paper's own equations or by a self-citation chain, to its own input.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces two new invariants (Sierpiński packing chromatic number and its upper version) as definitions, not as entities requiring independent evidence. No free parameters are fitted; axioms are standard results from the cited literature.

assumptions (3)
  • standard math Packing chromatic numbers of coronas P_n ⊙ 2K1, P_n ⊙ 3K1, and P_n ⊙ pK1 as given in Theorems 2.1-2.3 from [23]
    Used in Sections 4.1, 4.2, 4.3 to compute lower bounds for upper Sierpiński packing chromatic numbers.
  • standard math Subcubic trees have packing chromatic number at most 7 (Sloper's result as cited in [3, Theorem 2.18])
    Cited as context for Lemma 4.2's claimed bound on class T.
  • standard math The packing coloring decision problem is NP-complete on trees (Fiala and Golovach [11])
    Used as motivation in Section 4.

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Pith. "Pith review of On Sierpi\'{n}ski packing chromatic number and recognition of Sierpi\'{n}ski products." pith.science (2026). https://pith.science/paper/SSGON2DJ

@misc{pith2026250706730,
  author       = {Pith},
  title        = {Pith review of: On Sierpi\'nski packing chromatic number and recognition of Sierpi\'nski products},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SSGON2DJ}},
  note         = {Machine review of arXiv:2507.06730}
}
abstract

The Sierpi\'{n}ski product $G \otimes _f H$ of graphs $G$ and $H$ with respect to a function $f \colon V(G)\rightarrow V(H)$ has the vertex set $V(G)\times V(H)$. For every $g\in V(G)$ it contains a disjoint copy $gH$ of $H$, and for every edge $gg'$ of $G$ there is the edge $(g,f(g'))(g',f(g))$ between $gH$ and $g'H$. In this paper, the Sierpi\'{n}ski packing chromatic number is defined as the minimum of $\chi_{\rho}(G\otimes _f H)$ over all functions $f$, where $\chi_{\rho}(X)$ is the packing chromatic number of $X$. The upper Sierpi\'{n}ski packing chromatic number is analogously defined as the maximum corresponding value. The (upper) Sierpi\'{n}ski packing chromatic number is determined for all Sierpi\'{n}ski product graphs whose both factors are complete. Sierpi\'{n}ski product graphs whose factors are paths or stars are also studied. Their Sierpi\'{n}ski packing chromatic number is always $3$, while their upper Sierpi\'{n}ski packing chromatic number is bounded from below and above. It is also proved that for a given graph $G$, it can be checked in polynomial time whether $G$ has a representation as a Sierpi\'{n}ski product graphs both factors of which being trees.

Figures

Figures reproduced from arXiv: 2507.06730 by the authors.

Figure 1
Figure 1. The Sierpi´nski product K5 ⊗f K4, where f(u1) = 2, f(u2) = 4, f(u3) = 4, f(u4) = 1, f(u5) = 3. We start with a couple of lemmas. Lemma 3.1 If m, n ≥ 2 and f ∈ KKm n , then diam(Km ⊗f Kn) = 3. Proof. Let G = Km ⊗f Kn. By the definition of the Sierpi´nski product, there is a connecting edge between any pair of fibers since Km is complete. In addition, 5 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. The Sierpi´nski product Gm1,m2 = K8 ⊗f K2, where f(ui) = 1 for i ∈ [5], and f(ui) = 2 otherwise. Consider now the representation Gm1,m2 of Km⊗fK2 with m1 and m2 as described above. We may without loss of generality assume that m1 ≥ m2. We infer that 9 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. The Sierpi´nski product P14 ⊗g K1,3. Color all leaves and all vertices of degree 2 of Pm ⊗g K1,n with color 1, all vertices (ui , 1) where g(ui) = 1 with color 3, and all the remaining vertices with color 2, see [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

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