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REVIEW 3 major objections 6 minor 2 references

How many ways to color the map of America?

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper counts the proper four-colorings of real country maps, reporting a formula for interlocking wheels and two conflicting values for the contiguous United States.

desk verdict A promising interlocking-wheel chromatic polynomial, but the paper's headline USA count is internally inconsistent and the main proof has unverified computer algebra gaps. read the letter →

arxiv 1908.05694 v1 pith:3HOVDBRR submitted 2019-08-15 math.HO math.CO

Rebekah Bassett , Jennifer Canizales , Jasbir S. Chahal , Thomas Fackrell , Vanessa Rico This is my paper · ORCID
classification math.HOmath.CO MSC 05-0205C30
keywords chromaticpolynomialmapcoloringFourColorTheoreminterlockingwheelsdeletion-contractionreductiongeographicgraphUSA
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 asks a neglected question: exactly how many proper colorings does an actual country's map have? It computes the answer for Canada (576 with three colors), for the twelve contiguous regions of France (5,184 with four colors), and for the lower 48 United States, where it reports 12,811,729,152 in one decomposition and 12,811,591,729,152 in another without acknowledging that the two differ. The route is a chromatic-polynomial computation that breaks map graphs into overlapping wheels, supported by a new formula for interlocking wheels.

What carries the argument

The machinery is the chromatic polynomial $\chi(G,t)$, combined with CRT-1 (overlap in a complete graph: $\chi(G,t)=\chi(G_1,t)\chi(G_2,t)/\chi(K_l,t)$) and CRT-2 (deletion-contraction: $\chi(G,t)=\chi(G-e,t)-\chi(G/e,t)$). The new object is the interlocking wheel $W_m\wedge_2 W_n$, two wheels identified along a two-vertex wedge; the Main Theorem formula expresses its chromatic polynomial in terms of wheel polynomials, letting the authors decompose large geographic graphs into pieces small enough for computer algebra.

What would settle it

Compute $\chi(W_6\wedge_2 W_7,4)$ directly with an independent deletion-contraction routine and compare it with the Main Theorem formula; if they differ, the interlocking-wheel formula and the France and USA counts are unsupported. Also rerun the paper's two decompositions of $G_A$ in a computer algebra system and compare the two reported values, 12,811,729,152 and 12,811,591,729,152.

Watch

Extended reading notes

Core claim

The authors claim that the chromatic polynomials of the Canada, France, and USA map graphs can be assembled from paths, cycles, complete graphs, and interlocking wheels using two reduction rules—deletion-contraction and overlap division—and that evaluating these polynomials at the appropriate number of colors gives the counts. The Main Theorem states a closed formula for $\chi(W_m\wedge_2 W_n,t)$, the chromatic polynomial of two wheels that share a two-vertex wedge, and this formula is what makes France and the USA computations feasible. For the USA the paper gives two values, 12,811,729,152 and 12,811,591,729,152, asserting both as the number of proper colorings from a four-color palette.

Load-bearing premise

The calculation assumes that two large algebraic expressions, which the paper says a computer program showed to be equal, really are equal; the paper shows no program code or output to verify this, and the two USA counts reported later disagree.

Editorial extensions

If this is right

  • If the Main Theorem is correct, the number of proper four-colorings of the lower 48 states is either 12,811,729,152 or 12,811,591,729,152 depending on which of the paper's two decompositions is trusted, and including Alaska and Hawaii multiplies the count by 16.
  • The count for the twelve contiguous regions of France is 5,184, and Canada's three-color count is 576.
  • The Main Theorem extends the known wheel chromatic-polynomial formula to pairwise wedge-overlapping wheels, so other country maps containing interlocking wheels can be treated by the same reduction.
  • The computed USA polynomial satisfies the paper's listed necessary conditions: it is monic of degree 48, has $t^{47}$ coefficient $-105$, has alternating signs, has zero constant term, and has coefficients summing to zero.

Reading between the lines

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

  • The two reported USA values differ by roughly a factor of a thousand, so at most one can be right; an independent rerun of either 48-vertex decomposition in a symbolic computer system would settle the count.
  • The Main Theorem's induction adds one wheel at a time, so the formula likely extends to chains of more than two interlocking wheels; testing a three-wheel chain would be a natural check.
  • The same deletion-contraction plus overlap-division recipe could produce chromatic polynomials for other real maps whose graphs contain interlocking wheels, such as Mexico, Brazil, or Germany's Länder.
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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

3 major / 6 minor

Summary. The paper computes chromatic polynomials of real geographic maps: Canada (576 three-colorings), metropolitan France (5184 four-colorings), and the contiguous United States. The USA computation is performed twice via deletion-contraction and CRT-1 decompositions of the 48-state graph G_A, and the paper presents a Main Theorem giving chi(W_m ^2 W_n, t) for interlocking wheels, used in the France and USA calculations. The paper's central claim is a single integer for the number of proper 4-colorings of the lower-48 map.

Significance. If correct, the Main Theorem would be a useful addition to the chromatic-polynomial toolbox for planar graphs containing interlocking wheels, and the Canada/France counts are natural data points not previously found in the literature. The paper uses only standard deletion-contraction identities and contains no fitted parameters, and it honestly describes the computational difficulties. However, the central USA count is given as two different integers with no reconciliation, and the crucial algebraic identities in the Main Theorem proof are asserted without machine-checkable support. As it stands, the headline result is not established.

major comments (3)
  1. [§6.3 and §7] The paper gives two different values for chi(G_A,4): in §6.3 (first attempt) it states 12,811,729,152, and in §7 (second attempt) it states 12,811,591,729,152. The second value is roughly 1000 times the first, and the text never acknowledges or reconciles the difference. Since the paper's central claim is a single integer for the number of proper 4-colorings of the lower-48 map, this internal contradiction means the main result is not established as written.
  2. [§5, Main Theorem proof] The two induction leaps in the proof of equation (3) are justified only by 'By Wolfram Mathematica e1-u2212e2 = 0', with no code, notebook, output, or independent algebraic derivation provided. These identities are load-bearing: formula (3) is used directly in the France computation and indirectly in the USA computations. Without a checkable verification, the Main Theorem and all subsequent counts rest on an unsupported assertion.
  3. [§6.3 and §7, second attempt] The second USA computation relies on Mathematica's evaluation of chi(X,t) for a 41-vertex, 93-edge graph X, but no code, output, or repoducible data are given for this step. Combined with the First Attempt, the reader has no way to verify either the decomposition into X and P or the final integers, and the claimed equality of the two attempts is not demonstrated.
minor comments (6)
  1. [§2] In the definition of P_2(V), the word 'carnality' should be 'cardinality'.
  2. [§5, proof of Theorem 3] The displayed expression 't(t−n)^{n−1}' appears to be a typographical corruption of t(t−1)^n; the intended algebra is clear from context but should be corrected.
  3. [§6.2, France polynomial] The final term of the displayed chromatic polynomial for G_F is '−3696y', which should be '−3696t'.
  4. [§6.3] In the footnote for the US map, 'Retrived' should be 'Retrieved'.
  5. [§5, Main Theorem base case] In the n=m=5 base case, the notation 'χ((W5∧2W5)+e,t)' is confusing because it does not name the graph before edge addition; the paragraph should define G=(W5∧2W5)−e explicitly before applying CRT-2.
  6. [§5, proof of Theorem 5] The sentence 'Let n≥3 be any integer such that it is true for all graphs with n≥3' is misworded; the intended inductive hypothesis is that the claim holds for all graphs on n vertices.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's counts derive from deletion-contraction and CRT-1/CRT-2; the two divergent USA values and unverified Mathematica steps are correctness risks, not circular reductions.

full rationale

The paper's derivation chain is self-contained. Section 5 proves the wheel and cycle chromatic polynomials from the two Chromatic Reduction Theorems, and Section 6 applies these identities to geographic graphs. The Main Theorem for interlocking wheels is proved by double induction; the only questionable step is the invocation 'By Wolfram Mathematica e1−e2 = 0' in the induction, but that is an unsupported computational assertion, not a self-referential definition, a fitted input renamed as a prediction, or a result imported from the authors' own prior work. No parameter is fitted, no load-bearing self-citation occurs, and no equation in the paper defines its target in terms of itself. The internal inconsistency in Section 7—where χ(GA,4) is first reported as 12,811,729,152 and later as 12,811,591,729,152—is a genuine correctness failure in the central claim, as is the lack of verifiable evidence for the Mathematica equalities. However, under the stated rules, correctness gaps and unsupported computations are not circularity. Therefore the appropriate circularity score is 0.

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

No fitted parameters; the chromatic polynomials are computed exactly via deletion-contraction. The only non-standard axiom is the unverified correctness of the Mathematica simplifications in the Main Theorem proof.

assumptions (3)
  • standard math Chromatic Reduction Theorems (CRT-1 and CRT-2)
    Used throughout Sections 4-7; standard deletion-contraction and overlap formulas, proved informally in the paper.
  • ad hoc to paper Mathematica's polynomial simplifications are correct
    The induction proof of the Main Theorem asserts 'By Wolfram Mathematica e1-e2 = 0' without showing the computation or providing code.
  • domain assumption Map graphs match geographic adjacencies
    The graphs for France and the USA are read off figures (Figures 9 and 16) with no edge lists supplied.

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Cite this review

Pith. "Pith review of How many ways to color the map of America?." pith.science (2026). https://pith.science/paper/3HOVDBRR

@misc{pith2026190805694,
  author       = {Pith},
  title        = {Pith review of: How many ways to color the map of America?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3HOVDBRR}},
  note         = {Machine review of arXiv:1908.05694}
}
read the original abstract

Although the Four Color Conjecture originated in cartography, surprisingly, there is nothing in the literature on the number of ways to color an actual geographic map with four or fewer colors. In this paper, we compute these numbers, with exponentially increasing order of difficulty, for the maps of Canada, France, and the USA. Our attempts to compute the latter two lead to some new results on the chromatic polynomial of graphs.

Figures

Figures reproduced from arXiv: 1908.05694 by the authors.

Figure 1
Figure 1. Cycles 3. Complete graph. G = Kn Here |V (G)| = n and E(G) = P2(V ), i.e. every two vertices u, v of G are adjacent. 4. Wheel. Wn on n vertices (n ≥ 4) [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Wheels Two graphs G1 = (V1, E1) and G2 = (V2, E2) are isomorphic, written as G1 ∼= G2, if there is a bijection f : V1 −→ V2 such that {u, v} ∈ E1 ⇐⇒ {f(u), f(v)} ∈ E2. For example, P2 ∼= K2 and K4 ∼= W4. 5. Real Life Graphs. By these we mean the graphs associated to, among others, the geographic entities such as the map of the USA. In this graph, the vertices are the states of the USA and two states are adjacent if … view at source ↗
Figure 3
Figure 3. The broken wheel W0 6 and W6/e Now suppose that G1 and G2 are two graphs on disjoint sets of vertices and both G1 and G2 contain subgraphs that are isomorphic to a complete graph Kl . A graph G is an overlap of G1 and G2 in Kl if it is obtained by identifying the subgraphs of G1 and G2 that are isomorphic to Kl . Example 2 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: Non-Isomorphic overlaps 5 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: Map of Canada [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: The graph GC of the map of Canada 1Map of Canada retrieved from: https://www.conceptdraw.com/How-To-Guide/geo-map-canada-prince-edward-island 8 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: The graph of K The graph K is an overlap of C4 in K2 and a subgraph which is a series of repeated overlaps of C3 in K2. By CRT-1, Theorem 2 and Theorem 3, χ(GC, t) = χ(T,t)χ(K,t) χ(K1,t) . But χ(K, t) = χ(C4,t)χ(C3,t) 3 χ(K2,t) 3 . Therefore, χ(GC, t) = t(t − 1)6 (t − …
Figure 8
Figure 8. Figure 8: Map GF of France 2Map of France retrieved from: http://evasion-online.com/tag/carte-des-regions-de-france-2017 9 [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: The graph GF of the 12 contiguous regions of France The graph GF contains not only cycles, but also wheels. Computing χ(GF , t) would have been equally easy if they overlapped in complete graphs. Instead the wheels ”interlock” in a double wedge ∧2 ( [PITH_FULL_IMAGE:f…
Figure 10
Figure 10. Figure 10: Double wedge ∧2 [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: The interlocking wheel W6 ∧2 W7 10 [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]
Figure 12
Figure 12. Figure 12: Delete and mod out surgery on W6 ∧2 W6 One could add the edge e = {1, 3} to make it complete and apply CRT-2, but when we mod out edges, in the process we end up with graphs that are in general not even planar. Thus computing χ(GF , t) requires some new ideas. To over…
Figure 13
Figure 13. Figure 13: Graph of G1 and G2 12 [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]
Figure 14
Figure 14. Figure 14: Interlocking wheels as a subgraph of GF Back to France. Now we can compute χ(GF , t). First note that W6 ∧2 W7 overlaps repeatedly with three 3-cycles in K2 to produce GF . 16 [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]
Figure 15
Figure 15. Figure 15: Map A, contains 48 states of the USA 17 [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]
Figure 16
Figure 16. Figure 16: The graph GA with 48 vertices and 105 edges We associate the graph GA of the contiguous 48 States of America to the geographic map of the USA3 , by placing a vertex on every state, putting an edge between two vertices if the corresponding states share a border (at mor…
Figure 17
Figure 17. Figure 17: Attempt 1 Using CRT-1, χ(Y, t) = χ(C3, t) 3 χ(K2, t) 2 . We calculated χ(N, t) similarly. Finally, by a repeated application of CRT-1, we get χ(GA, t) = χ(W, t)χ(Y, t)χ(N, t)χ(C3, t) 3 χ(K2, t) 4χ(K1, t) (4) Then using Mathematica to compute χ(W, t), and plugging it i…
Figure 18
Figure 18. Figure 18: Attempt 2 We see that χ(GA, t) = χ(X, t)χ(P, t) χ(K2, t) Surprisingly, Mathematica is able to compute the chromatic polynomial for X, a graph with 41 vertices and 93 edges. And of course, using methods previously discussed, χ(P, t) can be calculated by applying repeat…

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

  1. [1]

    Harary, Combinatorics, Addison-Wesley (1969)

    F. Harary, Combinatorics, Addison-Wesley (1969)

    work page 1969

  2. [2]

    Read, An Introduction to Chromatic Polynomials , J

    R.C. Read, An Introduction to Chromatic Polynomials , J. Combinatorial Theory, 4(1968), 52–71. 22

    work page 1968

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