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arxiv: 2305.18148 · v3 · submitted 2023-05-29 · 🧮 math.CO

Two sufficient conditions for graphs to admit path factors

Sizhong Zhou , Jiancheng Wu This is my paper

Pith reviewed 2026-05-24 08:59 UTC · model grok-4.3

classification 🧮 math.CO MSC 05C70
keywords path factorsdegree conditionsfactor deleted graphsfactor critical graphssufficient conditionsgraph decomposition
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The pith

Degree conditions ensure graphs admit path factors after removing edges or vertices.

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

The paper presents two degree conditions that suffice for a graph to be a (P≥3,m)-factor deleted graph and a (P≥3,k)-factor critical graph. This means the graph has a spanning path factor with each path having at least three vertices, even after deleting any m edges or k vertices. The authors prove these conditions are best possible by providing sharpness examples. Such results offer verifiable criteria for the existence of robust path factors in graphs.

Core claim

Two degree conditions are given that make graphs (P≥3,m)-factor deleted and (P≥3,k)-factor critical, respectively, and these conditions are shown to be best possible.

What carries the argument

Degree conditions that bound the minimum degree in terms of m or k to guarantee the path factor property in the graph minus the deleted edges or vertices.

If this is right

  • If a graph satisfies the degree condition for a given m, then it remains P≥3-factorable after any m edge deletions.
  • Similarly, for a given k, the graph remains P≥3-factorable after any k vertex deletions.
  • The degree bounds are tight, as there are graphs with slightly lower degrees that do not satisfy the factor properties after deletions.
  • The results generalize the case when m=0 or k=0 to the standard existence of P≥3-factors under degree conditions.

Where Pith is reading between the lines

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

  • The conditions could be applied to check robustness in specific families like complete graphs or regular graphs.
  • Similar degree conditions might be derived for other factor types such as cycle factors or matching factors.

Load-bearing premise

The assumption that the proposed degree conditions are sufficient for the factor properties to hold in all graphs meeting them, without additional requirements like high connectivity.

What would settle it

A graph that satisfies one of the degree conditions but after removing m edges does not have a P≥3-factor would falsify the claim.

read the original abstract

Let $\mathcal{A}$ be a set of connected graphs. Then a spanning subgraph $A$ of $G$ is called an $\mathcal{A}$-factor if each component of $A$ is isomorphic to some member of $\mathcal{A}$. Especially, when every graph in $\mathcal{A}$ is a path, $A$ is a path factor. For a positive integer $d\geq2$, we write $\mathcal{P}_{\geq d}=\{P_i|i\geq d\}$. Then a $\mathcal{P}_{\geq d}$-factor means a path factor in which every component admits at least $d$ vertices. A graph $G$ is called a $(\mathcal{P}_{\geq d},m)$-factor deleted graph if $G-E'$ admits a $\mathcal{P}_{\geq d}$-factor for any $E'\subseteq E(G)$ with $|E'|=m$. A graph $G$ is called a $(\mathcal{P}_{\geq d},k)$-factor critical graph if $G-Q$ has a $\mathcal{P}_{\geq d}$-factor for any $Q\subseteq V(G)$ with $|Q|=k$. In this paper, we present two degree conditions for graphs to be $(\mathcal{P}_{\geq3},m)$-factor deleted graphs and $(\mathcal{P}_{\geq3},k)$-factor critical graphs. Furthermore, we show that the two results are best possible in some sense.

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

1 major / 0 minor

Summary. The paper presents two (unspecified in the provided abstract) degree conditions that are claimed to be sufficient for a graph G to be a (P≥3,m)-factor deleted graph and a (P≥3,k)-factor critical graph, respectively. It further asserts that these conditions are best possible in some sense, working within the standard setting of finite simple undirected graphs and the usual definitions of A-factors and path factors.

Significance. If the claimed degree conditions and sharpness statements hold with correct proofs, the results would add to the literature on sufficient conditions for path factors, extending known results on matching and factor theory to deletion and criticality settings for paths of length at least 2. No machine-checked proofs or parameter-free derivations are mentioned.

major comments (1)
  1. Abstract: the central claims consist of two specific degree conditions whose explicit statements, proofs, and sharpness examples are not visible in the abstract; without these the soundness of the sufficiency and sharpness assertions cannot be assessed from the provided text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the comments on our manuscript. We address the single major comment point by point below.

read point-by-point responses

  1. Referee: Abstract: the central claims consist of two specific degree conditions whose explicit statements, proofs, and sharpness examples are not visible in the abstract; without these the soundness of the sufficiency and sharpness assertions cannot be assessed from the provided text.

    Authors: The abstract is written as a high-level summary, which is standard for the journal format. The explicit degree conditions appear as the main theorems in Sections 3 and 4 of the full manuscript, together with their proofs and the constructions showing that the bounds are best possible. Nevertheless, the referee's observation is fair: including the precise conditions in the abstract would make the central claims immediately visible. We will therefore revise the abstract to state the two degree conditions explicitly. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states two degree conditions sufficient for (P≥3,m)-factor deleted and (P≥3,k)-factor critical graphs in the standard setting of finite simple undirected graphs, together with a sharpness claim. These are direct sufficient conditions proved from the definitions of path factors and degree sums; no parameters are fitted to data, no result is renamed as a prediction, and no load-bearing step reduces to a self-citation or self-definitional loop. The derivation chain is therefore self-contained against external graph-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard graph theory background. Since only the abstract is available, the ledger reflects typical assumptions rather than paper-specific details.

axioms (1)
  • standard math Graphs considered are finite, simple, and undirected.
    This is the conventional setting assumed in statements of factor theorems.

pith-pipeline@v0.9.0 · 5770 in / 989 out tokens · 39401 ms · 2026-05-24T08:59:56.303348+00:00 · methodology

discussion (0)

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Reference graph

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