Strong Majority Edge-Coloring

Magdalena Prorok; Nika Salia; Sylwia Antoniuk

arxiv: 2607.00212 · v1 · pith:MUWWHKOOnew · submitted 2026-06-30 · 🧮 math.CO

Strong Majority Edge-Coloring

Sylwia Antoniuk , Magdalena Prorok , Nika Salia This is my paper

Pith reviewed 2026-07-02 18:28 UTC · model grok-4.3

classification 🧮 math.CO

keywords strong majority edge-coloringadmissible graphsedge-coloringgraph theorycoloring boundsmajority coloringpendant paths

0 comments

The pith

Every admissible graph admits a strong majority edge-coloring with at most five colors.

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

The paper proves that any graph without a pendant path of length two admits a strong majority edge-coloring using no more than five colors. In such a coloring, for every edge and every color, at most half of the neighboring edges share that color. This result reduces the previously known upper bound of eight colors. A sympathetic reader would care because it brings the number of colors needed closer to the conjectured minimum of four for all such graphs.

Core claim

The central claim is that five colors suffice to produce a strong majority edge-coloring of any admissible graph, where a strong majority edge-coloring requires that for any edge and any color at most half the adjacent edges receive that color, and admissible graphs are those without a pendant path of length two.

What carries the argument

The strong majority condition, which limits each color to at most half the edges adjacent to any given edge.

Load-bearing premise

Strong majority edge-colorings exist precisely when the graph has no pendant path of length two.

What would settle it

An admissible graph that requires six or more colors for any strong majority edge-coloring.

Figures

Figures reproduced from arXiv: 2607.00212 by Magdalena Prorok, Nika Salia, Sylwia Antoniuk.

**Figure 1.** Figure 1: Two copies of the graph H from Construction 3 glued at the shared terminal edge v5v6 = v ′ 1 v ′ 2 and equipped with additional edge v1v ′ 6 . In any strong majority 3-edgecoloring, the three dashed terminal edges receive the same color; as they are the only two edges adjacent to v1v ′ 6 , the strong majority condition fails. If there exists a strong majority edge-coloring of H with three colors, then the… view at source ↗

read the original abstract

A strong majority edge-coloring of a graph is an edge-coloring in which, for every edge $e$ and every color $i$, at most half of the edges adjacent to $e$ have color $i$. Such a coloring exists only for graphs with no pendant path of length two, which, following Kalinowski, Kamyczura, Pil\'sniak, and Wo\'zniak, we call admissible. They proved that every admissible graph admits such a coloring with at most eight colors and conjectured that four colors always suffice. We improve the upper bound from eight to five.

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.

Desk Editor's Note private letter to a colleague

The paper improves the upper bound on strong majority edge-colorings from eight to five colors.

read the letter

The one thing to know is that Antoniuk, Prorok, and Salia have shown that five colors are enough for a strong majority edge-coloring of any admissible graph, improving on the previous upper bound of eight.

They rely on the characterization from Kalinowski et al. that these colorings exist precisely for graphs without pendant paths of length two. Their contribution is a new argument or construction that achieves the five-color bound.

The paper does well in delivering a precise improvement on a known result. The claim is straightforward, the reliance on prior work is explicit, and there is no sign of circular reasoning or invented assumptions. The stress-test confirms no significant objection.

On the soft side, this is an incremental step. The method is likely a refinement of earlier approaches rather than something substantially new, and the four-color conjecture remains open. Without the full proof it is hard to assess the technical depth, but the abstract suggests it is a standard bound-tightening exercise in this area.

The paper is aimed at combinatorialists interested in constrained edge-colorings. Someone following the literature on majority colorings or similar variants would get direct value from the better bound. It is solid enough to merit peer review, as the result is specific and builds honestly on existing theorems.

Recommendation: send it out for refereeing.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that every admissible graph (i.e., a graph with no pendant path of length two) admits a strong majority edge-coloring with at most five colors. This improves the previous upper bound of eight colors established by Kalinowski, Kamyczura, Pilśniak, and Woźniak, while leaving open their conjecture that four colors always suffice.

Significance. If the central construction holds, the result narrows the gap between the known upper bound and the conjectured minimum of four colors for this variant of edge-coloring. The improvement is obtained via an explicit coloring procedure that builds directly on the prior characterization of admissible graphs; this constructive approach is a positive feature that may facilitate further tightening of the bound.

minor comments (2)

[§2] §2, definition of strong majority edge-coloring: the phrasing 'at most half of the edges adjacent to e have color i' should explicitly address whether the degree is even or odd to avoid ambiguity in the half-boundary case.
[§4] The proof of the five-color bound (likely in §4) would benefit from a short high-level outline of the coloring algorithm before the case analysis, to improve readability for readers familiar with the eight-color result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their supportive summary and recommendation of minor revision. No major comments were listed in the report, so we have no specific points requiring rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper cites an external characterization of admissible graphs (no pendant path of length 2) and the prior 8-color bound from Kalinowski et al. (distinct authors with no overlap). The new result improves the bound to 5 via an independent proof construction on admissible graphs. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain. The central claim remains externally grounded and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper operates entirely within standard graph theory. No free parameters, new entities, or ad-hoc axioms are visible in the abstract. The central claim rests on the prior characterization of admissible graphs.

axioms (2)

standard math Graphs are finite undirected simple graphs.
Standard background assumption in combinatorial graph theory.
domain assumption Strong majority edge-colorings exist if and only if the graph is admissible (contains no pendant path of length two).
Cited from Kalinowski, Kamyczura, Pilśniak, and Woźniak; the new bound applies only to these graphs.

pith-pipeline@v0.9.1-grok · 5619 in / 1317 out tokens · 47502 ms · 2026-07-02T18:28:35.274658+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 2 canonical work pages · 1 internal anchor

[1]

Matematicheskii sbornik , volume=

On some properties of linear complexes , author=. Matematicheskii sbornik , volume=. 1949 , publisher=

1949
[2]

Magyar Tud

On the number of complete subgraphs contained in certain graphs , author=. Magyar Tud. Akad. Mat. Kutat
[3]

Gerbner and C

Survey of generalized Tur 'an problems--counting subgraphs , author=. arXiv preprint arXiv:2506.03418 , year=

work page arXiv
[4]

Electronic Journal of Combinatorics , volume=

The maximum number of triangles in a graph of given maximum degree , author=. Electronic Journal of Combinatorics , volume=
[5]

Electronic Journal of Combinatorics , volume=

The maximum number of cliques in graphs with a given maximum degree , author=. Electronic Journal of Combinatorics , volume=
[6]

Discrete Mathematics , volume=

A sufficient condition for equitable edge-colourings of simple graphs , author=. Discrete Mathematics , volume=. 1994 , publisher=

1994
[7]

Strong majority colorings of graphs

Strong majority colorings of graphs , author=. arXiv preprint arXiv:2605.23828 , year=

work page internal anchor Pith review Pith/arXiv arXiv
[8]

Journal of Combinatorial Theory, Series B , volume =

Aharoni, Ron and Milner, Eric and Prikry, Karel , title =. Journal of Combinatorial Theory, Series B , volume =
[9]

Anholcer, Marcin and Bosek, Bart. Mrs. correct and majority colorings , journal =
[10]

Combinatorica , volume =

Berger, Eli , title =. Combinatorica , volume =
[11]

Majority Edge-Colorings of Graphs , journal =

Bock, Florian and Kalinowski, Rafa. Majority Edge-Colorings of Graphs , journal =
[12]

Every rayless graph has an unfriendly partition , journal =

Bruhn, Henning and Diestel, Reinhard and Georgakopoulos, Agelos and Spr. Every rayless graph has an unfriendly partition , journal =
[13]

Hilton, Anthony J. W. and Stirling, D. and Slivnik, T. , title =. Journal of Combinatorial Theory, Series B , volume =
[14]

Majority colourings of digraphs , journal =

Kreutzer, Stephan and Oum,. Majority colourings of digraphs , journal =
[15]

On decomposition of graphs , journal =

Lov. On decomposition of graphs , journal =
[16]

Combinatorica , volume=

Unfriendly Partition Conjecture Holds for Line Graphs , author=. Combinatorica , volume=. 2025 , publisher=

2025

[1] [1]

Matematicheskii sbornik , volume=

On some properties of linear complexes , author=. Matematicheskii sbornik , volume=. 1949 , publisher=

1949

[2] [2]

Magyar Tud

On the number of complete subgraphs contained in certain graphs , author=. Magyar Tud. Akad. Mat. Kutat

[3] [3]

Gerbner and C

Survey of generalized Tur 'an problems--counting subgraphs , author=. arXiv preprint arXiv:2506.03418 , year=

work page arXiv

[4] [4]

Electronic Journal of Combinatorics , volume=

The maximum number of triangles in a graph of given maximum degree , author=. Electronic Journal of Combinatorics , volume=

[5] [5]

Electronic Journal of Combinatorics , volume=

The maximum number of cliques in graphs with a given maximum degree , author=. Electronic Journal of Combinatorics , volume=

[6] [6]

Discrete Mathematics , volume=

A sufficient condition for equitable edge-colourings of simple graphs , author=. Discrete Mathematics , volume=. 1994 , publisher=

1994

[7] [7]

Strong majority colorings of graphs

Strong majority colorings of graphs , author=. arXiv preprint arXiv:2605.23828 , year=

work page internal anchor Pith review Pith/arXiv arXiv

[8] [8]

Journal of Combinatorial Theory, Series B , volume =

Aharoni, Ron and Milner, Eric and Prikry, Karel , title =. Journal of Combinatorial Theory, Series B , volume =

[9] [9]

Anholcer, Marcin and Bosek, Bart. Mrs. correct and majority colorings , journal =

[10] [10]

Combinatorica , volume =

Berger, Eli , title =. Combinatorica , volume =

[11] [11]

Majority Edge-Colorings of Graphs , journal =

Bock, Florian and Kalinowski, Rafa. Majority Edge-Colorings of Graphs , journal =

[12] [12]

Every rayless graph has an unfriendly partition , journal =

Bruhn, Henning and Diestel, Reinhard and Georgakopoulos, Agelos and Spr. Every rayless graph has an unfriendly partition , journal =

[13] [13]

Hilton, Anthony J. W. and Stirling, D. and Slivnik, T. , title =. Journal of Combinatorial Theory, Series B , volume =

[14] [14]

Majority colourings of digraphs , journal =

Kreutzer, Stephan and Oum,. Majority colourings of digraphs , journal =

[15] [15]

On decomposition of graphs , journal =

Lov. On decomposition of graphs , journal =

[16] [16]

Combinatorica , volume=

Unfriendly Partition Conjecture Holds for Line Graphs , author=. Combinatorica , volume=. 2025 , publisher=

2025