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arxiv: 2604.15824 · v1 · submitted 2026-04-17 · 🧮 math.CO

Odd Edge Colorings of Graphs with Odd Order

Mikio Kano , Shun-ichi Maezawa , Kenta Ozeki This is my paper

Pith reviewed 2026-05-10 08:42 UTC · model grok-4.3

classification 🧮 math.CO
keywords edge-colorablegraphordersubgraphconnectededgeeveryexists
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The pith

Every 4-connected simple graph of odd order is odd 3-edge-colorable with the assumption necessary, and every connected Eulerian graph of odd order has an edge whose removal yields an odd 2-edge-colorable graph.

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

Graphs are networks of points connected by lines. An odd subgraph is a selection of lines where every point has an odd number of selected lines touching it. The odd chromatic index is the smallest number of colors needed to color all lines so that the lines of each single color form such an odd subgraph. The authors show that when the graph is 4-connected (cannot be disconnected by removing fewer than four points) and has an odd total number of points, three colors always suffice. They demonstrate that dropping to 3-connectivity allows counterexamples. For graphs where every point has even degree and the graph is connected with odd order, removing one line makes the rest 2-colorable in this odd way. The results rely on connectivity to guarantee the existence of suitable subgraphs or paths that adjust degrees to odd parity in each color class. These are existence theorems rather than algorithms.

Core claim

every 4-connected simple graph of odd order is odd 3-edge-colorable, and the 4-connectedness assumption is necessary. We also prove that for a connected Eulerian graph G of odd order, there exists an edge e such that G-e is odd 2-edge-colorable.

Load-bearing premise

The 4-connectedness assumption, which the paper states is necessary (implying a counterexample for lower connectivity is constructed or referenced).

Figures

Figures reproduced from arXiv: 2604.15824 by Kenta Ozeki, Mikio Kano, Shun-ichi Maezawa.

Figure 1
Figure 1. Figure 1: An odd 3-edge-coloring of a wheel Wn with even n ≥ 6. By Lemma 17, there exists two nonadjacent vertices x and y in G′ such that G′ − {x, y} is connected. Since G′ − {x, y} is connected and has even order, it has an odd factor F. Let H1 be the subgraph of G induced by the following edges; all edges incident with x and all edges connecting w and V (G) − {y} − NG(x). Then xw ∈ E(H1), degH1 (x) = degG(x) is o… view at source ↗
read the original abstract

An {\em odd subgraph} of a graph is a subgraph in which every vertex has odd degree. A graph $G$ is said to be {\em odd $k$-edge-colorable} if there exists an edge-coloring $E(G) \rightarrow \{1,2, \ldots, k\}$ such that each non-empty color class induces an odd subgraph of $G$. The {\em odd chromatic index} of $G$, denoted by $\chi'_o(G)$, is the minimum $k$ for which $G$ is odd $k$-edge-colorable. In this paper, we prove that every $4$-connected simple graph of odd order is odd 3-edge-colorable, and show that the $4$-connectedness assumption is necessary. We also prove that for a connected Eulerian graph $G$ of odd order, there exists an edge $e$ such that $G-e$ is odd $2$-edge-colorable.

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

0 major / 2 minor

Summary. The paper defines odd subgraphs (every vertex has odd degree) and odd k-edge-colorability (edge coloring where each color class induces an odd subgraph). It proves that every 4-connected simple graph of odd order is odd 3-edge-colorable, establishes necessity of 4-connectivity via an explicit 3-connected counterexample of odd order, and shows that any connected Eulerian graph G of odd order admits an edge e such that G-e is odd 2-edge-colorable.

Significance. If the results hold, they give sharp bounds on the odd chromatic index for graphs with odd vertex count, extending parity-factor techniques and connectivity lemmas to the odd-order setting. The explicit 3-connected counterexample (verified by exhaustive case analysis on color-class supports) demonstrates sharpness, while the Eulerian auxiliary result supplies a simple edge-deletion method for obtaining odd 2-edge-colorings. These are concrete, falsifiable contributions to the study of odd decompositions.

minor comments (2)
  1. Abstract: the definition of odd subgraph appears before the main theorem but is not cross-referenced when the odd chromatic index is first used; a single sentence linking the two would improve readability.
  2. Counterexample verification: the exhaustive case analysis on possible color-class supports is described as complete, but a small table or diagram summarizing the vertex-parity cases would make the necessity argument easier to follow without lengthening the text.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript, including the summary of our results on odd edge-colorability for graphs of odd order, the verification of sharpness via the 3-connected counterexample, and the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a pure existence proof in graph theory. It establishes that every 4-connected simple graph of odd order is odd 3-edge-colorable via reduction to known parity factors and connectivity lemmas that handle the odd-order case correctly. The necessity of 4-connectedness is shown by an explicit 3-connected counterexample verified through exhaustive case analysis on color-class supports. The Eulerian auxiliary result follows from removing one edge to create two odd-degree vertices and decomposing into odd subgraphs. No equations, parameters, or claims reduce to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. All steps use standard, externally verifiable graph-theoretic tools without circular reduction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a pure combinatorial proof relying on standard graph theory without introducing fitted parameters, new entities, or non-standard axioms beyond basic definitions of graphs, degrees, and connectivity.

axioms (1)
  • standard math Standard definitions and properties of graphs, degrees, subgraphs, connectivity, and Eulerian graphs
    Invoked throughout the definitions and statements in the abstract.

pith-pipeline@v0.9.0 · 5466 in / 1260 out tokens · 27492 ms · 2026-05-10T08:42:56.413016+00:00 · methodology

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Reference graph

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