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arxiv: 1907.11600 · v1 · pith:3Z772IMUnew · submitted 2019-07-26 · 🧮 math.CO

Edge-partitioning 3-edge-connected graphs into paths

Tereza Klimo\v{s}ov\'a , St\'ephan Thomass\'e This is my paper

Pith reviewed 2026-05-24 15:28 UTC · model grok-4.3

classification 🧮 math.CO
keywords edge-partitionpaths of fixed length3-edge-connected graphsminimum degreegraph decompositionedge-disjoint pathsconnectivity conditions
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The pith

For every path length l, 3-edge-connected graphs with high enough minimum degree admit an edge-partition into paths of length l when the edge count is divisible by l.

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

The paper proves that fixing any path length l allows a threshold d_l on minimum degree such that 3-edge-connected graphs meeting or exceeding that degree can have their edges split into paths each exactly l long. This holds whenever the total number of edges is a multiple of l. The result lowers the required edge-connectivity from 24 down to 3 while keeping the minimum-degree condition. It also shows that dropping to 2-edge-connectivity allows counterexamples even at high degree, making the connectivity assumption sharp.

Core claim

For every positive integer l there exists an integer d_l such that every 3-edge-connected graph with minimum degree at least d_l whose number of edges is divisible by l can be edge-partitioned into paths of length l.

What carries the argument

3-edge-connectivity together with a minimum-degree threshold depending on l, which together guarantee the existence of the path partition.

If this is right

  • The edge-partition exists for arbitrarily long paths once the degree threshold is met.
  • The connectivity requirement cannot be relaxed to 2-edge-connectivity.
  • The result applies uniformly to all l once the corresponding d_l is chosen.
  • Earlier theorems requiring 24-edge-connectivity are improved by replacing that condition with 3-edge-connectivity plus a larger degree bound.

Where Pith is reading between the lines

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

  • The same approach might adapt to decompositions into cycles of fixed length under additional parity conditions.
  • Explicit bounds on d_l could be extracted from the proof to make the result effective for small l.
  • The decomposition could be useful for constructing regular subgraphs or for load-balancing in networks modeled by such graphs.

Load-bearing premise

The graph must satisfy 3-edge-connectivity, because the partition can fail for some 2-edge-connected graphs no matter how large the minimum degree becomes.

What would settle it

A 3-edge-connected graph whose minimum degree exceeds d_l, whose edge count is divisible by l, yet whose edges cannot be partitioned into paths of length l.

Figures

Figures reproduced from arXiv: 1907.11600 by St\'ephan Thomass\'e, Tereza Klimo\v{s}ov\'a.

Figure 1
Figure 1. Figure 1: Examples of complex paths with three wiggly edges and one ordinary [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Replacing an edge e by a gadget He. Proof. Let L be greater than L2 obtained from Theorem 30 applied to r and c as in our hypothesis. Let G∗ be the complex graph obtained from G by replacing each wiggly edge e with endpoints v1,v2 by a gadget He consisting of a clique of size L + 1 and unit wiggly edges e1, e2 with endpoints v1, v0 2 and v1, 0 v2, respectively, where v 0 1 and v 0 2 are (arbitrary) distinc… view at source ↗
Figure 3
Figure 3. Figure 3: Replacing edges between a and B. Indices at vertices in B ”transfer” to the new edge or hyperedge. index of the original hyperedge at that vertex. (We do not define the lengths and indices at a; they become irrelevant in the next step of the construction.) Since we contracted A and Gi was 3-edge-connected, the resulting complex hypergraph is 3-edge-connected. Then, we repeatedly replace pairs of edges inci… view at source ↗
Figure 4
Figure 4. Figure 4: The reducing procedure. (Stubs are depicted as arrows.) [PITH_FULL_IMAGE:figures/full_fig_p037_4.png] view at source ↗
read the original abstract

We show that for every l, there exists d_l such that every 3-edge-connected graph with minimum degree d_l can be edge-partitioned into paths of length l (provided that its number of edges is divisible by l). This improves a result asserting that 24-edge-connectivity and high minimum degree provides such a partition. This is best possible as 3-edge-connectivity cannot be replaced by 2-edge connectivity.

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 manuscript proves that for every positive integer l there exists d_l such that every 3-edge-connected graph G with minimum degree at least d_l and |E(G)| divisible by l admits an edge-decomposition into paths of exact length l. The result improves a prior theorem that required 24-edge-connectivity (instead of 3-edge-connectivity) together with high minimum degree, and the authors observe that the connectivity hypothesis is sharp because the statement fails for 2-edge-connected graphs.

Significance. If correct, the theorem substantially lowers the connectivity threshold for path decompositions of this form and establishes that 3-edge-connectivity is the right minimal assumption. The explicit divisibility proviso and the existential quantification over d_l are standard in the area; the sharpness statement with respect to 2-edge-connectivity is a useful clarification.

minor comments (2)
  1. [Abstract] The abstract and introduction should state the range of l explicitly (positive integers) and confirm whether the result holds for l=1 (trivial) and l=2.
  2. [Introduction] Notation for the path length l and the threshold d_l should be introduced consistently in the first paragraph of the introduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. The report accurately captures the main result and its improvement over the 24-edge-connectivity threshold, as well as the sharpness with respect to 2-edge-connectivity. No specific major comments are listed in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states an existential theorem in graph theory: for each fixed l there is a d_l such that every 3-edge-connected graph of minimum degree at least d_l admits an edge-partition into paths of length l (when the edge count is divisible by l). This is proved from first principles in combinatorial graph theory and improves an earlier result that required 24-edge-connectivity; the improvement is obtained by a direct argument rather than by re-using any fitted parameter, self-defined quantity, or load-bearing self-citation whose validity rests inside the present manuscript. No equation or claim reduces to its own input by construction, and the divisibility proviso is stated explicitly as an external hypothesis. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract only; no explicit free parameters, invented entities, or non-standard axioms are stated. The claim rests on standard definitions of edge-connectivity and minimum degree in finite graphs.

axioms (1)
  • standard math Standard definitions and basic properties of finite undirected graphs, edge-connectivity, and minimum degree.
    The theorem is phrased entirely in these classical terms.

pith-pipeline@v0.9.0 · 5595 in / 1162 out tokens · 28636 ms · 2026-05-24T15:28:53.721698+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 1 internal anchor

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