On List Equitable Total Colorings of the Generalized Theta Graph

Jeffrey A. Mudrock; Max Marsh; Tim Wagstrom

REVIEW 1 major objections 4 minor 43 references

On List Equitable Total Colorings of the Generalized Theta Graph

T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that for every generalized theta graph $G = \Theta(l_1, \ldots, l_m)$, the total graph $T(G)$ is equitably $k$-choosable for every $k \geq m+2$, confirming the List Equitable Total Coloring Conjecture for this family.

desk verdict Extends the LETCC program to new families, but a false claim in Lemma 22 leaves Theorem 7 unproven; worth a careful revision rather than desk rejection. read the letter →

arxiv 1908.01657 v1 pith:P35SX7R4 submitted 2019-08-02 math.CO

Jeffrey A. Mudrock , Max Marsh , Tim Wagstrom This is my paper · ORCID

classification math.CO MSC 05C15

keywords graphcoloringtotalequitablelistchoosabilitygeneralizedthetasubdivisionofstars

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

This paper tries to establish the List Equitable Total Coloring Conjecture for a broad family of graphs: every generalized $\theta$ graph, meaning two vertices joined by internally disjoint paths, and also every subdivision of a star. Equitable $k$-choosability is a strong list-coloring requirement: from any assignment of $k$ allowed colors per vertex of the total graph, one must color adjacent and incident elements with distinct colors while no color appears more than $\lceil N/k\rceil$ times. The conjecture says the total graph of any simple graph is equitably $k$-choosable once $k$ is at least the maximum of its list chromatic number and half its maximum degree plus two. The authors prove the threshold $k \geq m+2$ works for a $\theta$ graph with $m$ paths, and $k \geq m+1$ works for a subdivision of a star, both sharp or nearly sharp. The result matters because it verifies the conjecture on a natural class of graphs and supports the general conjecture.

What carries the argument

The central machinery is the reduction $T(G) = [\Theta(2l_1, \ldots, 2l_m)]^2$ together with a greedy extension lemma, Lemma 10, adapted from Kostochka, Pelsmajer, and West. Lemma 10 says that if a graph has an equitable coloring after deleting a set $S$ of vertices, and each vertex of $S$, ordered suitably, has few remaining neighbors, then the coloring extends to an equitable coloring of the whole graph. Nearly every proof in the paper uses this lemma, with carefully chosen deleted sets $S$ so that the leftover graph has maximum degree at most $4$ and can be colored by the Kierstead-Kostochka theorem, or is a path square or cycle square colored by known propositions. The critical $k = m+2$ case is completed by case-specific lemmas that color three vertical layers of the $\theta$ graph in such a way that every color class remains small.

What would settle it

Run Case (2) of Lemma 22 on $\Theta(2,4,4,4)^2$ with the stated greedy order $w, u, v_{4,1}, v_{3,2}, v_{4,2}$ and search over 6-assignments satisfying the case's hypothesis: if any such list assignment forces a four-color coloring of $S_1$ whose repeated pair is $v_{4,1}/v_{3,2}$ or $v_{3,2}/v_{4,2}$ while the color $c$ already occurs in the coloring of $G-S_1$, the contradiction claimed in the lemma does not follow. Exhibiting one such assignment would show that the proof of Theorem 7 does not establish the lemma at $m=3$; finding none would support the lemma.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is Theorem 7: if $G = \Theta(l_1, \ldots, l_m)$, then $T(G)$, the total graph, is equitably $k$-choosable for every $k \geq m+2$. Since a total graph can be represented as the square of the doubly subdivided graph, the proof actually works with squares of generalized $\theta$ graphs, proving stronger intermediate bounds $k \geq m+3$ when the paths are long enough and then handling the critical case $k = m+2$ by a sequence of lemmas, including special cases $\Theta(2,4,4)$, $\Theta(2,4,4,4)$, and general constructions for $\Theta(2,4,\ldots,4)$, $\Theta(4,\ldots,4)$, and $\Theta(l_1,l_2,\ldots,l_m)$ with $l_m \geq 6$. For $m=1$ and $m=2$, the result reduces to known choosability of path squares and cycle squares. Together with the star-subdivision result, this verifies the List Equitable Total Coloring Conjecture for these graphs.

Load-bearing premise

The proof's most delicate load-bearing step is the assertion in Lemma 22's case 2 that a greedy four-color coloring of the five-vertex set $S_1$ must put the repeated color on $w$ and $v_{4,1}$; if other repeat patterns are possible, the contradiction used to finish that case does not follow.

Editorial extensions

If this is right

Every generalized theta graph satisfies the List Equitable Total Coloring Conjecture, since the theorem supplies equitable $k$-choosability for every $k \geq m+2$, the conjecture's required threshold for these graphs.
Subdivisions of stars are equitably $k$-choosable for $k \geq m+1$, one color below the conjecture's general threshold, so those graphs are handled with room to spare.
The threshold $m+2$ cannot be lowered to $m+1$ for $m=1$ or $m=2$, because the relevant path squares and cycle squares are known failures at $m+1$; whether $m+1$ works for $m \geq 3$ is left open as Question 8.
Together with earlier work on stars, double stars, and trees of maximum degree three, the conjecture now holds for a larger family of graphs built from internally disjoint paths.
The greedy extension lemma gives a reusable template: any graph whose square has low maximum degree after deleting a small structured set can be attacked by the same argument.

Reading between the lines

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

Editorial: the same square-of-subdivision reduction suggests the conjecture may be approachable for other graphs obtained by subdividing edges of low-maximum-degree graphs, not just stars and theta graphs.
Editorial: a natural testable extension is Question 8; a computer search over small theta graphs with $m=3,4$ could quickly check whether $m+1$ choosability fails and guide a proof or counterexample.
Editorial: the proof's special small cases, $\Theta(2,4,4)$ and $\Theta(2,4,4,4)$, indicate that the difficult behavior is concentrated in short paths; understanding those cases may be the key to settling the open threshold.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper adds two new families to the List Equitable Total Coloring Conjecture program: subdivisions of stars (Theorem 6) and generalized theta graphs (Theorem 7). The work is honest and mostly careful. Theorem 6 looks solid; the induction in Section 2 is clean and the use of known results (Kierstead-Kostochka, path-square choosability) is appropriate. The theta graph part is the genuinely new contribution.

The soft spot is Lemma 22, and it is load-bearing. In case (2), after coloring G−S1 with seven distinct colors, the authors claim that any greedy coloring of S1 using exactly four colors must repeat the color of w on v4,1. That is false. With the stated order (w, u, v4,1, v3,2, v4,2), the repeated pair can instead be v4,1/v3,2 or v3,2/v4,2. The contradiction that follows relies on (V(G)−S1) ⊆ N(w) ∪ N(v4,1), but that inclusion does not help when the repeated color lies on v4,1/v3,2 or v3,2/v4,2—that color could already be used by h on a vertex outside both neighborhoods, such as v2,2. So the proof that no color is used three times fails for the k=m+2 case with m=4 and lengths (2,4,4,4). Theorem 7 as written is incomplete.

There is a smaller issue in Lemma 24: the bound |L′(w)| ≥ m+1 is overstated. In that setting w’s neighbors in S1 are only u and v1,2, so the correct bound is m. Since the argument colors S2 with m+1 distinct colors, the weaker bound does not automatically give an SDR. This is likely repairable but needs checking.

None of this is a desk-reject situation. The paper is not incoherent, the gap is localized, and the rest of the arguments are in good shape. It deserves a serious referee. If the authors can fix Lemma 22—by a different choice of S1, a more careful greedy argument, or an additional case—the theta graph result would be a useful, citable contribution to the LETCC program.

Recommendation: send to peer review, but flag Lemma 22 as a required revision.

Referee Report

1 major / 4 minor

Summary. The paper studies list equitable total colorings of total graphs. Theorem 6 states that if G is a subdivision of the star K_{1,m}, then T(G) is equitably k-choosable for every k ≥ m+1 (with k ≥ 3 when m=1). Theorem 7 states that if G is a generalized theta graph Θ(l_1,...,l_m), then T(G) is equitably k-choosable for every k ≥ m+2, thereby verifying the List Equitable Total Coloring Conjecture for this family. Since T(G) is isomorphic to the square of the graph obtained by doubling all path lengths, the proofs are carried out on squares of generalized theta graphs. The paper uses a general extension lemma (Lemma 10) together with known equitable choosability results, and then provides ad hoc arguments for the critical case k=m+2, including Lemmas 21 through 26 for special length vectors.

Significance. If the proof were complete, Theorems 6 and 7 would be a substantial contribution: Theorem 6 is stronger than the LETCC for subdivisions of stars, and Theorem 7 would verify the LETCC for a new infinite family of graphs. The general framework of Lemmas 10 through 20 is clean and applies previous results appropriately. The treatment of the many small cases is explicit and constructive. However, a load-bearing step in the k=m+2 case is not proved correctly as written, so the main theorem is not established in the present version.

major comments (1)

[Section 3, Lemma 22] In the proof of Lemma 22, case (2), the assertion "By the way h' is constructed, it must be that h'(w)=h'(v_{4,1})=c" is false. In G[S_1] the only non-adjacent pairs are {w,v_{4,1}}, {v_{4,1},v_{3,2}}, and {v_{3,2},v_{4,2}}. With the stated greedy order w,u,v_{4,1},v_{3,2},v_{4,2} and the lower bounds |L'(v_{4,1})|≥2, |L'(v_{3,2})|≥4, |L'(v_{4,2})|≥5, a proper 4-color output can repeat v_{4,1} and v_{3,2} (for example, assign colors 1,2,3,3,4 in that order) or can repeat v_{3,2} and v_{4,2} (colors 1,2,3,4,4). The subsequent contradiction uses (V(G)-S_1) ⊆ N(w)∪N(v_{4,1}), which only excludes a repeated color c that appears on a vertex adjacent to w or v_{4,1}. If the repeated pair is v_{4,1}/v_{3,2} or v_{3,2}/v_{4,2}, the color c may already be used by h on a vertex such as v_{2,2} or v_{2,3}, which is not adjacent to v_{3,2}; no contradiction follows. Consequently the proof does not establish that no color is used more than twice, so the k=6 case for [Θ(2,4,4,4)]^2, and therefore the m=4 case of Theorem 7 with original lengths (1,2,2,2), remains unproved.

minor comments (4)

[Section 3, Lemma 21] In the proof of Lemma 21, case (2), the definition L'(v_{2,i}) = L(v_{2,i}) - {f(v) : v ∈ N_G(v_{3,i}) - S_1} should use N_G(v_{2,i}) - S_1; the displayed lower bounds |L'(v_{2,1})|≥2, |L'(v_{2,2})|≥3, and |L'(v_{2,3})|≥2 are computed from the latter neighborhood. As written, the bound for v_{2,2} would only be |L'(v_{2,2})|≥1 and the subsequent coloring of G[S_1] would not be guaranteed proper.
[Section 3, Lemma 26] In the verification of the neighborhood bounds, |N_G(v_{m,1}) - S| is 2, not 1, because v_{m,1} is adjacent in G-S to both v_{1,1} and v_{2,1}. The required inequality still holds since k - m = 2, but the displayed value should be corrected.
[Section 3, Lemma 20] After setting x_1 = v_{m,3}, x_2 = v_{1,1}, x_3 = u, x_{k-1} = v_{m,2}, and x_k = v_{m,1}, the sentence naming the remaining vertices should begin with x_4, not x_3, since x_3 has already been assigned.
[Section 3, Lemma 22] The existence of the coloring h of G-S_1 using seven distinct colors is plausible but not justified in the text; it follows from Hall's theorem because each list has size 6 and the set {L(v) : v ∈ V(G)-S_1} contains at least two distinct lists, forcing the union of all seven lists to have size at least 7. A brief justification would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation is self-contained apart from independently published choosability theorems used as black boxes.

full rationale

I walked the proof chain of Theorems 6 and 7 and found no step in which an output is defined in terms of the target result, no fitted quantity is relabeled as a prediction, and no load-bearing argument reduces to a self-citation. The new results are proved by explicit coloring constructions: the star-subdivision case reduces to Lemmas 10-16, and the generalized theta case reduces to Lemmas 19-26, each of which builds colorings from Lemma 10 and from independent results such as the Kierstead-Kostochka theorem (Theorem 4) and the published path-square and cycle-square choosability results cited as Propositions 13 and 17 from [18]. Although [18] shares an author with the present paper, the specific propositions used are stated and proved for path powers and cycle powers, not for subdivisions of stars or generalized theta graphs; they are applied to proper subgraphs or base cases and do not contain the target theorem. The LETCC is introduced as a conjecture from [18] and then verified, not assumed. No ansatz is smuggled in through citation, no uniqueness theorem is imported from the authors' prior work, and no known result is merely renamed in new coordinates. The reviewer-identified concern about the greedy assertion in Lemma 22 is a potential correctness gap in a proof step, not an equivalence of the proof's conclusion with its inputs, so under the hard rules it does not constitute circularity and does not raise the circularity score.

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

No free parameters or invented entities. The paper depends on two external theorems: the Kierstead-Kostochka bound and the path/cycle power choosability results from [18]. These are standard published results, not assumptions tailored to this paper.

assumptions (3)

standard math Theorem 4 of Kierstead and Kostochka (2013) provides equitable list colorings for graphs with maximum degree up to 7 using Δ+1 colors.
Used to color G-S when Δ(G-S) ≤ 4 in Lemmas 11, 12, 20, 25, and 26.
standard math Proposition 13 of Kaul, Mudrock, and Pelsmajer (2018): powers of paths are equitably k-choosable for k ≥ p+1.
Used to color path squares in Lemmas 14, 15, and for the cases m=1,2 in Theorem 6.
standard math Proposition 17 of Kaul, Mudrock, and Pelsmajer (2018): powers of cycles are equitably k-choosable for k ≥ 2p when n ≥ 2p+2.
Used for the case m=2 in Theorem 7.

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Pith. "Pith review of On List Equitable Total Colorings of the Generalized Theta Graph." pith.science (2026). https://pith.science/paper/P35SX7R4

@misc{pith2026190801657,
  author       = {Pith},
  title        = {Pith review of: On List Equitable Total Colorings of the Generalized Theta Graph},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P35SX7R4}},
  note         = {Machine review of arXiv:1908.01657}
}

abstract

In 2003 Kostochka, Pelsmajer, and West introduced a list analogue of equitable coloring called equitable choosability. A $k$-assignment, $L$, for a graph $G$ assigns a list, $L(v)$, of $k$ available colors to each $v \in V(G)$, and an equitable $L$-coloring of $G$ is a proper coloring, $f$, of $G$ such that $f(v) \in L(v)$ for each $v \in V(G)$ and each color class of $f$ has size at most $\lceil |V(G)|/k \rceil$. In 2018, Kaul, Mudrock, and Pelsmajer subsequently introduced the List Equitable Total Coloring Conjecture which states that if $T$ is a total graph of some simple graph, then $T$ is equitably $k$-choosable for each $k \geq \max \{\chi_\ell(T), \Delta(T)/2 + 2 \}$ where $\Delta(T)$ is the maximum degree of a vertex in $T$ and $\chi_\ell(T)$ is the list chromatic number of $T$. In this paper we verify the List Equitable Total Coloring Conjecture for subdivisions of stars and the generalized theta graph.

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Behzad, Graphs and their chromatic numbers, Ph.D

M. Behzad, Graphs and their chromatic numbers, Ph.D. Thesis, M ichigan State University, 1965

work page 1965

[2] [2]

Beier, J

J. Beier, J. Fierson, R. Haas, H. M. Russel, K. Shavo, Classifying coloring graphs, Discrete Mathematics 339 (2016), no. 8, 2100-2112

work page 2016

[3] [3]

J. I. Brown, C. A. Hickman, A. D. Sokal. D. G. Wagner, On the chr omatic roots of generalized theta graphs Journal of Combinatorial Theory Series B 83 (2001), 272-297

work page 2001

[4] [4]

J. M. Carraher, T. Mahoney, G. J. Puelo, D. B. West, Sum-paint ability of generalized theta- graphs, Graphs and Combinatorics , 31 (2015), no. 5, 1325-1334

work page 2015

[5] [5]

B. -L. Chen, K. -W. Lih, P. -L. Wu, Equitable coloring and the maxim um degree, Eur. J. Combin. 15 (1994), 443-447

work page 1994

[6] [6]

Chunling, L

T. Chunling, L. Xiaohui, Y. Yuansheng, L. Zhihe, Equitable total c oloring of Cm□ Cn, Disc. App. Math. 157 (2009), 596-601

work page 2009

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A. Dong, J. Wu, Equitable coloring and equitable choosability of plan ar graphs without chordal 4- and 6-cycles, arXiv: 1806.01064 (preprint), 2018

work page Pith review arXiv 2018

[8] [8]

A. Dong, X. Zhang, Equitable coloring and equitable choosability of graphs with small maximum average degree, Discussiones Mathematicae Graph Theory 38 (2018), 829-839

work page 2018