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Local Weak Degeneracy of Planar Graphs

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Planar graphs are weakly degenerate under local girth bounds

desk verdict A significant conjecture-resolution with a real gap: Observation 2.6, used throughout the induction, is false as stated, though the proof is likely patchable. read the letter →

arxiv 2504.21821 v1 pith:7QT4YZOU submitted 2025-04-30 math.CO

Ewan Davies , Evelyne Smith-Roberge This is my paper · ORCID
classification math.CO MSC 05C1505C10
keywords weakdegeneracyplanargraphslocalgirthcorrespondencecolouringDP-colouringlistcanvasDelSaveoperation
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 proves a local-girth analogue of weak degeneracy for planar graphs. For each vertex $v$, define its local girth $g(v)$ as the length of a shortest cycle through $v$; the theorem says that whenever a function $f$ gives each vertex at least $\max\{7-g(v),2\}$ units, every planar graph admits a legal deletion sequence using the Del and DelSave operations. This is strictly stronger than ordinary degeneracy, and it directly implies the conjectured correspondence-colouring (DP-colouring) version of the local girth list colouring theorem. A sympathetic reader should care because one statement simultaneously explains the three classical pillars of planar list colouring: 5-list-colourability, 4-list-colourability when triangles are absent, and 3-list-colourability when girth is at least five. The proof works by establishing a much more detailed inductive statement about 'canvases' with exactly three small exceptional shapes.

What carries the argument

The load-bearing object is the canvas: a tuple $(G,P,A,B,f)$ in which $G$ is a plane graph, $P$ is an acceptable path or cycle on the outer face (a path of at most four vertices with a girth condition on its internal vertices), $A$ and $B$ are independent sets on the outer face boundary outside $P$ consisting of vertices of girth at least five (with $f=1$) and girth three (with $f=2$), and $f$ satisfies the local girth lower bounds (C4a)--(C4e). The induction removes not just a few vertices but an arbitrarily long path $R$ along the outer face, chosen according to eight case rules (R1)--(R8) that depend on the values of $f$; the corresponding deletion sequences (D1)--(D8) simulate a coordinated colour choice using Del and DelSave. Two structural chord lemmas classify every 2- and 3-chord in a minimum counterexample as belonging to one of the three exceptional canvas types, which is what guarantees that after a path removal the remaining smaller canvas still satisfies the induction hypothesis unless exactly one of $A$ or $B$ loses independence. Those two failure modes are then handled in Sections 4 and 5 by deleting the offending vertices and checking that the resulting smaller canvas is unexceptional.

What would settle it

Conduct a brute-force search over all plane graphs with at most ten vertices for a canvas $(G,P,A,B,f)$ satisfying Definition 2.5 with no (X1)--(X3) structure, for which the associated function $f_K(v)=f(v)-|N(v)\cap P|$ admits no legal Del/DelSave deletion sequence of $G-P$. Such a canvas would directly falsify Theorem 2.8 and hence the main theorem. Alternatively, exhibit a planar graph $G$ and the canonical local girth function $f(v)=\max\{7-g(v),2\}$ with no legal weak-degeneracy sequence; Theorem 1.11 says none exists.

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Extended reading notes

Core claim

The central discovery is Theorem 1.11: if $G$ is planar and $f$ is a local girth function, meaning $f(v)\ge \max\{7-g(v),2\}$ for every vertex $v$, then $G$ is weakly $f$-degenerate. In the paper's language, weak $f$-degeneracy means there is a legal sequence of operations $\mathrm{Del}(v)$ and $\mathrm{DelSave}(v,w)$ that removes every vertex of $G$, where Del decrements all neighbours of $v$ and DelSave decrements all neighbours except the chosen neighbour $w$, with legality requiring $f(v)>f(w)$ for a save. The theorem is proved through the more restrictive canvas statement (Theorem 2.8): every canvas $(G,P,A,B,f)$ — a plane graph with an acceptable path or cycle $P$ on the outer face, independent sets $A$ and $B$ of vertices with extra-restricted values, and specified lower bounds on $f$ — is weakly $f$-degenerate unless it exhibits one of three exceptional configurations (X1), (X2), or (X3). Taking $P$ to be a single edge on the outer face recovers Theorem 1.11. Since weak $f$-degeneracy with a function $f$ implies $(f(v)+1)$-correspondence colouring, the theorem resolves the conjecture that every planar graph is local girth correspondence colourable, and it does so without any infinite family of exceptional structures.

Load-bearing premise

The load-bearing premise is the chord classification inside a minimum counterexample: Lemmas 3.12 and 3.13 assert that every 2-chord and every 3-chord whose endpoints are not internal vertices of $P$ forces one of the three exceptional canvas structures (X1), (X2), or (X3); if a chord of any other shape appeared, the induction step that removes a long boundary path could not conclude that the reduced canvas is still unexceptional.

Editorial extensions

If this is right

  • Theorem 1.11 implies Theorem 1.7: every planar graph is local girth correspondence colourable, settling the open conjecture in that direction.
  • Because weak degeneracy is the more flexible parameter, the theorem also implies the analogous statement for correspondence-painting, the online version of correspondence colouring.
  • The theorem simultaneously recovers previously separate results: planar graphs are weakly 4-degenerate, planar graphs of girth at least five are weakly 2-degenerate, and the classical list-colouring theorems for planar graphs with girth 3, 4, and 5.
  • The proof does not yield an exponential lower bound on the number of colourings; the paper states this explicitly and proposes a stronger availability conjecture that, via Proposition 1.13, would give exponentially many local girth correspondence colourings.

Reading between the lines

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

  • A natural next step, not taken in the paper, is to check whether the chord-classification lemmas can be re-proved for other sparse graph classes; for locally planar graphs or graphs on surfaces, the same canvas framework could transfer if the exceptional structures are adjusted.
  • The eight-case path removal scheme suggests a computational experiment: enumerate all legal Del and DelSave sequences for small planar graphs and measure the maximum average availability; consistent values above 1 would support the paper's availability conjecture, while a graph whose every legal sequence has average availability exactly 1 would refute it.
  • Because the proof is a finite-case induction, it likely yields a polynomial-time algorithm that constructs the legal deletion order; if so, the correspondence colouring result becomes constructive, not just existential.
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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

2 major / 4 minor

Summary. The paper proves that every planar graph is weakly f-degenerate for every local girth function f, i.e. f(v) ≥ max{7−g(v),2}; as a corollary it proves Postle and Smith-Roberge's conjecture that every planar graph is local girth correspondence colourable. The proof is by minimal counterexample to a stronger induction statement (Theorem 2.8) formulated in terms of canvases with a distinguished outer-face path P and two independent sets A and B of vertices with restricted f-values. Sections 3–5 establish structural properties of a minimum counterexample, remove a path R from the outer face, and handle the two possible failures of the resulting reduced canvas. The argument is detailed and, apart from the gaps noted below, appears to be self-contained.

Significance. If the proof is correctly repaired, this is a strong and significant result: it simultaneously strengthens weak 4-degeneracy of planar graphs and weak 2-degeneracy of planar girth-5 graphs, resolves the correspondence-colouring conjecture of Postle and Smith-Roberge, and does so with a self-contained argument whose exceptions are three finite configurations rather than infinite families. The paper also honestly discusses why its methods do not yield exponential colouring bounds. These are substantial strengths. The main concern is technical; I believe it is addressable, but it affects the foundation of the induction.

major comments (2)
  1. [Observation 2.6 (Section 2)] Observation 2.6 is false as stated. Consider the plane graph with outer cycle b-u-v-w-b and chord u-w, and let P=vw, A=∅, B={b}, f(b)=2, f(u)=3, f(v)=f(w)=0. This is an unexceptional canvas: b has girth 3, u satisfies (C4d), and v and w lie in P. Deleting u leaves the path b-w-v, in which b has infinite girth, so (G-u,P,A,B,f) violates (C3) and is not a canvas. The observation is invoked at the start of Lemmas 3.4, 3.5, 3.6, 3.7, 3.8, 3.12 and 3.13, and similar reasoning is used in Sections 4 and 5; the induction steps as written are therefore not justified. A repair is plausible: after deleting vertices, redefine B as the subset of the old B that still has girth 3 in the smaller graph, since girth can only increase under vertex deletion and f=2 still satisfies (C4c). Because vertex deletion cannot create new adjacencies, exceptionality cannot newly arise. However, this repair must be stated explicitly and checked in every invocation; it is not a harmless rephrasing, since the classifications in Lemmas 3.12 and 3.13 depend on the exact membership of B in exceptions (X1) and (X3).
  2. [Sections 4 and 5] Several canvas checks in Sections 4 and 5 are asserted very tersely, for example 'it is easy to check that w∈\widehat B' in Claim 1 and the verification that the final operations in Section 5 are legal. Since the false Observation 2.6 is used elsewhere, these checks need to be written out in full under the corrected understanding of how B and \widehat B are updated after vertex deletion. The current level of detail is not sufficient for a proof whose main mechanism is repeated reduction to a smaller canvas.
minor comments (4)
  1. [Lemma 3.11] This lemma is stated without proof and is not used later in the paper. Either provide the proof or remove the statement, since an unproved lemma in a minimal-counterexample proof is a formal gap.
  2. [Section 2, after Definition 2.5] The text says that exceptional canvases are not weakly f-degenerate because fK assigns a negative value to some vertex. For type (X3), the defining adjacencies give fK(v1)=fK(v2)=0, so this justification is not accurate; if the sentence is only motivational, it should be reworded.
  3. [Section 3.2, cases (R1)–(R8)] The cases for defining R are described as lexicographically minimal without explicit mutually exclusive conditions. This is understandable, but later arguments depend on knowing exactly which case applies, so the cases should be stated with clean disjoint conditions.
  4. [Claim 1, Section 4] The step 'it is easy to check that w∈\widehat B' should be expanded: one should explicitly verify that w has girth 3 in \widehat G and that \widehat f(w)=2, especially since the proof of (C3) depends on this membership.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorem is established by an independent minimal-counterexample induction from definitions, and self-citations are contextual rather than load-bearing.

full rationale

The paper's central result, Theorem 1.11, is reduced to the stronger canvas statement Theorem 2.8, and Theorem 2.8 is proved by a minimum-counterexample induction over the canvas definition. The only imported result is Lemma 2.2 from Bernshteyn and Lee, a general monotonicity statement for weak degeneracy that does not encode the planar local-girth conclusion. The authors' self-citations ([4], [19], [20]) are used for context, inspiration, and the conjecture being resolved, not as black-box substitutes for the argument; the paper even states that its proof is self-contained and no longer relies on a theorem of Thomassen from [24]. No fitted parameter is renamed as a prediction, and no exceptional-case or chord classification is defined in terms of the theorem being proved. The skeptic's concern about Observation 2.6 is a potential correctness gap in an auxiliary assertion, not a circularity: the paper does not define the conclusion into the premise. Consequently, there is no circular step to report.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard graph theory and one imported general-purpose lemma; no fitted parameters or new physical entities are introduced. The canvas formalism is an internal proof device, not an independent postulate.

assumptions (4)
  • standard math Well-ordering principle: a minimum counterexample exists if any counterexample exists.
    Used to select the minimum canvas K in Section 3; standard foundation of induction.
  • standard math Eulerian plane embedding facts: the outer face boundary δG and chord separations behave as in plane topology.
    Definitions 2.3 and 3.2 and Lemmas 3.4 to 3.13 use standard properties of plane graphs; no deep external theorem is imported.
  • standard math Bernshteyn and Lee monotonicity of weak degeneracy (Lemma 2.2).
    Imported from [3]; it is used to compare functions f and to justify legality when f is increased. It is a general-purpose lemma, not a restatement of the target theorem.
  • domain assumption Finite simple graphs only.
    Stated in the introduction; the canvas definitions rely on simplicity, with no loops or parallel edges.

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Cite this review

Pith. "Pith review of Local Weak Degeneracy of Planar Graphs." pith.science (2026). https://pith.science/paper/7QT4YZOU

@misc{pith2026250421821,
  author       = {Pith},
  title        = {Pith review of: Local Weak Degeneracy of Planar Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7QT4YZOU}},
  note         = {Machine review of arXiv:2504.21821}
}
abstract

Thomassen showed that planar graphs are 5-list-colourable, and that planar graphs of girth at least five are 3-list-colourable. An easy degeneracy argument shows that planar graphs of girth at least four are 4-list-colourable. In 2022, Postle and Smith-Roberge proved a common strengthening of these three results: with $g(v)$ denoting the length of a shortest cycle containing a vertex $v$, they showed that if $G$ is a planar graph and $L$ a list assignment for $G$ where $|L(v)| \geq \max\{3,8-g(v)\}$ for all $v \in V(G)$, then $G$ is $L$-colourable. Moreover, they conjectured that an analogous theorem should hold for correspondence colouring. We prove this conjecture; in fact, our main theorem holds in the still more restrictive setting of weak degeneracy, and moreover acts as a joint strengthening of the fact that planar graphs are weakly 4-degenerate (originally due to Bernshteyn, Lee, and Smith-Roberge), and that planar graphs of girth at least five are weakly 2-degenerate (originally due to Han et al.).

Figures

Figures reproduced from arXiv: 2504.21821 by the authors.

Figure 1
Figure 1. The exceptional cases of Theorem 2.8. Vertices of the path P are squares. If a canvas contains the structure described by (X1), (X2), or (X3) above, we call it an excep￾tional canvas of type (X1), (X2), or (X3), respectively. See [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Cases for Lemma 3.8 in which we argue that the 1-chord uv (the thick edge) cannot be in a minimal counterexample. follows that K′ is not a counterexample to Theorem 2.8 and thus K′ is weakly f ′ -degenerate. But then by Lemma 2.2 the canvas K is weakly f-degenerate, a contradiction. In particular, Lemma 3.10 implies that δG does not contain three consecutive vertices v with f(v) = 4. Moreover, a nearly identical arg… view at source ↗
Figure 3
Figure 3. 2-chords xyz of the type that are not ruled out by Lemma 3.12. A nearly identical proof gives the following result. We show examples in [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: 3-chords xyzw of the type we cannot rule out in Lemma 3.13: g(y) ≥ 5, neither x nor w is an internal vertex of P, and one side of the chord is an exception of type (X2) or (X3). As a consequence of Lemmas 3.4, 3.8 and 3.9, we have the following fact in a minimum coun￾t…
Figure 5
Figure 5. Figure 5: The decomposition of δG into P P′P ′′uk. If v0 exists then v0 ∈ A, and we have f(v1) ≥ 2 in any case. The labels uk and vt+1 refer to the same vertex for convenience. we do not state mutually exclusive conditions and we remove R according to the lexicographically minim…
Figure 6
Figure 6. Figure 6: Necessary graph structure in the case that B˜ is not independent. The black vertices are removed by the sequence of operations σ and Gˆ = G − {v0, v1, v2, v3}. Note that it may be the case that v4 = uk (and hence v4 ∈ V (P)). Claim 1. Kˆ is a canvas and Bˆ \ B = {w}. P…
Figure 7
Figure 7. Figure 7: Structure relevant to the proof of Claim 1. . . . v1 v2 v3 . . . w v v ′ [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]
Figure 8
Figure 8. Figure 8: In the proof of Claim 1, an edge vv′ that cannot be induced by Aˆ as then the 3-chord v1vv′v3 violates Lemma 3.13. Thus, Kˆ is indeed a canvas. We argue that it is unexceptional. If Kˆ is an exceptional canvas of type (X1), then since K is unexceptional, since B˜ \ B =…
Figure 9
Figure 9. Figure 9: Necessary graph structure in the case that A˜ is not independent. The general approach to handle this case will be to remove w1 and w2 as well as R. The following claim is used to argue that it is possible to remove V (R) ∪ {w1, w2} from G − P via a sequence of legal o…
Figure 10
Figure 10. Figure 10: Showing that Kˆ cannot be an exceptional canvas of type (X3) in Claim 7. If u ∈ A and v ̸∈ B, then given g(u) ≥ 5, we have that uv together with the endpoint of P neighbouring v form a 2-chord violating Lemma 3.12. If u ̸∈ A and v ∈ B, then uv together with the endpoi…
Figure 11
Figure 11. Figure 11: Ruling out neighbours of w4 in R for Claim 9. The structures for case (ii) are similar. Let w3 and w4 be vertices as described in Claim 9, and let vi the neighbour of w4 in V (R). In particular, choose w3w4 to minimize i. If w3 is adjacent to w1, let Q = v1w1w3w4vi . …
Figure 12
Figure 12. Figure 12: The structure given by Claim 9 for an edge w3w4 ∈ Gˆ[Aˆ] chosen to minimize i. The edges of the path Q are bold. δG2 is the cycle v1 · · · viw4 · · · w1v1. Since G is a plane graph, our choice of i gives us the following. Observation 5.1. There does not exist a vertex…
Figure 13
Figure 13. Figure 13: Ruling out Q = v1w1w2w3w4vi and i = 4 in Claim 11. Let A′′ be the set of vertices in V (G′′) \ V (P) with f ′′(v) ≤ 1, and B′′, the set of vertices v ∈ V (G′′) \ V (P) with g(v) = 3 and f(v) = 2. Since v2v1 is a subpath of R, f ′′(v) ≥ ˜f(v) for all v ∈ V (G′′) \ {v3}…

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