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arxiv: 2507.04516 · v2 · submitted 2025-07-06 · 💻 cs.DS

The planar edge-coloring theorem of Vizing in O(nlog n) time

Patryk J\k{e}drzejczak , {\L}ukasz Kowalik This is my paper

Pith reviewed 2026-05-19 05:37 UTC · model grok-4.3

classification 💻 cs.DS
keywords planar graphsedge coloringVizing theoremgraph algorithmstime complexityrecoloringbounded genus
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The pith

Vizing's edge-coloring theorem for planar graphs with maximum degree 8 can be implemented in O(n log n) time

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

The paper establishes an algorithm that finds a proper edge coloring with 8 colors for any planar graph whose maximum degree is exactly 8, and the running time is O(n log n). It reaches this bound by adapting the recoloring steps in Vizing's original proof so they avoid the quadratic cost of a direct implementation. A sympathetic reader would care because the result finishes the algorithmic version of Vizing's planar theorem for every degree where the theorem guarantees a coloring exists.

Core claim

By modifying the recoloring procedure of Vizing, we obtain an algorithm for edge-coloring planar graphs of maximum degree 8 with 8 colors in O(n log n) time. This extends the earlier O(n log n) algorithm of Chrobak and Nishizeki, which applied only to graphs with maximum degree at least 9, and the new method generalizes to graphs of bounded genus.

What carries the argument

Modified Vizing recoloring procedure adapted for the degree-8 case to preserve the O(n log n) runtime bound established for higher degrees

If this is right

  • The O(n log n) time bound now holds for edge-coloring all planar graphs with Δ ≥ 8 using Δ colors
  • The same algorithmic approach applies directly to graphs embeddable on surfaces of bounded genus
  • Efficient computation of optimal edge colorings is now available for every planar graph covered by Vizing's theorem

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The modification technique may suggest ways to obtain similar runtime gains for other structural theorems about colorings on planar and near-planar graphs
  • The work shows that small adjustments to an existing proof structure can close the remaining efficiency gap without changing the underlying existence result

Load-bearing premise

The recoloring procedure from Vizing's proof can be modified for the Δ=8 case while preserving the O(n log n) runtime bound established for higher degrees

What would settle it

A planar graph with maximum degree 8 on which the modified recoloring procedure either fails to produce a valid 8-edge-coloring or requires more than O(n log n) time

Figures

Figures reproduced from arXiv: 2507.04516 by {\L}ukasz Kowalik, Patryk J\k{e}drzejczak.

Figure 2
Figure 2. Figure 2: Subgraph B2. Subgraph B is called a butterfly and edge xy is called butterfly-like. An edge is called reducible when it is weak or butterfly-like. Later, we will show how to reduce butterfly-like edges. Here, we prove that introducing them is sufficient for a linear bound for the number of reducible edges. Theorem 2. Let G = (V, E) be a planar graph without isolated vertices and such that ∆(G) ≤ 8. Then G … view at source ↗
Figure 3
Figure 3. Figure 3: Type 1ab. y a, b z x c ⊥ c a (a, b) (a, b) [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Type 3ab. y a, b x c v z b ⊥ a c (a, b) [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Type 5ab. y a, c, d x b v1 z b, d v2 ⊥ c a a c (a, b) (c, d) [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: Without loss of generality, assume π(xz) = a, π(xv1) = b, π(xv2) = c, π(xv3) = d, π¯(x) = {f}, where a, b, c, d, f are pairwise distinct. If f ∈ π¯(y), then π(xy) := f gives σ. Hence, we can assume f ∈ π(y). Assume f ∈ π¯(z). If a ∈ π¯(y), then π(xz) := f and π(xy) := a give σ. Hence, we can assume a ∈ π(y), which leaves us with two cases. In the first case, when π(yv1) = f and π(yv3) = a, we can obtain σ … view at source ↗
Figure 9
Figure 9. Figure 9: Proof of Lemma 7, the case when B is isomorphic to B1. 3. π(yv1) = f, π(zv2) = f. For each of these cases, we show how we can obtain σ in a few subcases. In Case 1, if π(zv2) ̸= b, we do swapπ(xv1, zv1) and π(xy) := b. If π(zv2) = b and π(yv1) ̸= c, we do swapπ(xv1, zv1), swapπ(xv2, zv2), and π(xy) := c. If π(zv2) = b and π(yv1) = c, we do swapπ(xv3, yv3), π(xz) := d, and π(xy) := a. In Case 2, if π(zv1) =… view at source ↗
Figure 10
Figure 10. Figure 10: Proof of Lemma 7, the case when B is isomorphic to B2. For any k ∈ π¯(y), we can assume there is an (f, k)-chain with endpoints x and y. Otherwise, e has type 1fk. Without loss of generality, assume π(xz) = a, π(xv1) = b, π(xv2) = c, and π(xv3) = d, where a, b, c, d, f are pairwise distinct. Assume π(yv4) ̸= a. If f ∈ π¯(z), setting π(xz) := f and π(xy) := a gives σ, so e has type 0. Hence, we can assume … view at source ↗
read the original abstract

In 1965, Vizing [Diskret. Analiz, 1965] showed that every planar graph of maximum degree $\Delta\ge 8$ can be edge-colored using $\Delta$ colors. The direct implementation of the Vizing's proof gives an algorithm that finds the coloring in $O(n^2)$ time for an $n$-vertex input graph. Chrobak and Nishizeki [J. Algorithms, 1990] have shown a more careful algorithm, which improves the time to $O(n\log n)$ time, though only for $\Delta\ge 9$. In this paper, we extend their ideas to get an algorithm also for the missing case $\Delta=8$. To this end, we modify the original recoloring procedure of Vizing. This generalizes to bounded genus graphs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims an O(n log n) time algorithm for Δ-edge-coloring planar graphs when Δ=8, achieved by modifying the recoloring step in Vizing's proof to close the gap left by Chrobak and Nishizeki's algorithm for Δ≥9. The approach is further generalized to graphs of bounded genus.

Significance. If the algorithmic result holds, it would represent a significant advance by providing an efficient implementation of Vizing's planar edge-coloring theorem for the remaining case of Δ=8. This builds upon established methods and offers a generalization that may have implications for topological graph algorithms. The work is notable for its focus on preserving the near-linear runtime through careful modification of the recoloring procedure.

major comments (2)
  1. Section 3 (modified recoloring procedure): the manuscript describes changes to Vizing's fan rotation and missing-color selection for Δ=8, but does not supply a new potential function or charging argument showing that alternating-path and fan depths remain bounded so that the Chrobak-Nishizeki O(log n) per-edge amortization is preserved under the tighter color budget.
  2. Section 4 (runtime analysis): the O(n log n) claim is asserted after the Δ=8 modification, yet no explicit bound is given on the size of searchable fans or recursive recoloring depth for degree-8 vertices that have only one missing color on both sides of a critical edge; this analysis is load-bearing for the central runtime guarantee.
minor comments (2)
  1. Abstract: the generalization to bounded-genus graphs is mentioned without stating the precise genus bound or any extra conditions required for the modification to carry over.
  2. Introduction: a short table comparing runtimes for Δ=8 versus Δ≥9 (and versus the quadratic direct implementation) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback on our manuscript. The comments correctly identify areas where the runtime analysis for the Δ=8 case requires additional explicit detail to fully substantiate the O(n log n) bound. We address each major comment below and will incorporate revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: Section 3 (modified recoloring procedure): the manuscript describes changes to Vizing's fan rotation and missing-color selection for Δ=8, but does not supply a new potential function or charging argument showing that alternating-path and fan depths remain bounded so that the Chrobak-Nishizeki O(log n) per-edge amortization is preserved under the tighter color budget.

    Authors: We agree that an explicit adaptation of the charging argument is needed to confirm the bounds under the modified recoloring for Δ=8. The manuscript relies on the fact that our missing-color selection rule for degree-8 vertices preserves the key invariant that each fan rotation or alternating-path extension can be charged against a distinct color deficiency, but this is not spelled out with a dedicated potential function. In the revised version we will add a new lemma in Section 3 that reuses the Chrobak-Nishizeki potential (with a minor adjustment for the single-missing-color case) and proves that the depth remains O(log n) because each step still increases the potential by Ω(1) while the total potential is bounded by O(n). revision: yes

  2. Referee: Section 4 (runtime analysis): the O(n log n) claim is asserted after the Δ=8 modification, yet no explicit bound is given on the size of searchable fans or recursive recoloring depth for degree-8 vertices that have only one missing color on both sides of a critical edge; this analysis is load-bearing for the central runtime guarantee.

    Authors: The referee is right that the current text does not supply a self-contained bound for the critical degree-8 case. While the overall amortization follows the same structure as Chrobak and Nishizeki, the manuscript does not explicitly verify that searchable fans remain O(log n) and that recursion depth stays logarithmic when both endpoints of a critical edge have only one missing color. We will add a dedicated paragraph and supporting lemma in Section 4 that derives these bounds directly from the modified recoloring rule, showing that the number of fan rotations per edge is still amortized O(log n) and that the recursion tree has logarithmic depth. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit modification of recoloring procedure with independent runtime argument

full rationale

The paper directly modifies Vizing's recoloring procedure to close the Δ=8 gap while extending the Chrobak-Nishizeki charging scheme for O(n log n) total work. No equation or parameter is defined in terms of the target runtime or coloring; the time bound is re-proven under the altered fan and path rules rather than assumed by construction. Prior work is cited from independent authors and used only as a starting point for the modification, with no self-citation chain or uniqueness theorem imported from the present authors. The derivation therefore remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard graph-theoretic background and prior algorithmic results rather than introducing new fitted parameters or postulated entities.

axioms (1)
  • domain assumption Vizing's theorem that every planar graph with maximum degree Delta >=8 is Delta-edge-colorable
    Invoked as the existence result whose algorithmic realization is being accelerated.

pith-pipeline@v0.9.0 · 5679 in / 1195 out tokens · 97111 ms · 2026-05-19T05:37:45.076415+00:00 · methodology

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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