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arxiv: 2605.00031 · v1 · submitted 2026-04-23 · 🧮 math.CO

Graphs with \{P₃,P₄,P₅\}-factors in terms of size and spectral radius

Zahoor Iqbal Bhat , S. Pirzada This is my paper

Pith reviewed 2026-05-09 21:09 UTC · model grok-4.3

classification 🧮 math.CO
keywords path factor{P3,P4,P5}-factorgraph sizeadjacency spectral radiusminimum degreespanning subgraphconnected graphfactorization
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The pith

A connected graph of order n with minimum degree δ has a {P3,P4,P5}-factor if its size or adjacency spectral radius is sufficiently large.

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

The authors prove sufficient conditions for a connected graph to contain a spanning subgraph whose components are paths of three, four or five vertices. One condition is a lower bound on the number of edges expressed in terms of n and the minimum degree δ. The other is a lower bound on the largest eigenvalue of the adjacency matrix. These criteria ensure that the graph can be covered completely by short paths, which is relevant for problems in graph decomposition and network partitioning. The proofs combine counting arguments with spectral techniques to verify the necessary conditions for such a factor.

Core claim

Let G be a connected graph of order n with minimum degree δ. There exists a function f(n, δ) such that if the number of edges in G is at least f(n, δ), then G has a {P3, P4, P5}-factor. There also exists a function g(n, δ) such that if the adjacency spectral radius of G is at least g(n, δ), then G has a {P3, P4, P5}-factor.

What carries the argument

The {P3,P4,P5}-factor, a spanning subgraph with every connected component being a path on three, four or five vertices, whose existence is guaranteed by lower bounds on graph size or adjacency spectral radius.

If this is right

  • If the edge count condition holds, the vertices of G can be partitioned into groups of size 3, 4 or 5 that can be connected by paths within the graph.
  • The spectral radius condition offers an algebraic test for the existence of the factor without explicitly constructing it.
  • These conditions generalize previous results on path factors by incorporating the minimum degree parameter.
  • Graphs meeting the thresholds cannot be too sparse or have small eigenvalues, preventing certain obstructions to path factors.

Where Pith is reading between the lines

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

  • The results suggest that for denser graphs the factor always exists, potentially leading to asymptotic versions as n grows.
  • Similar spectral conditions might apply to other types of factors, such as cycles or trees of bounded size.
  • These bounds could be used in heuristic algorithms for path cover problems in large networks.

Load-bearing premise

The size and spectral radius thresholds are large enough to ensure that the graph satisfies any Tutte-like condition required for the existence of the {P3,P4,P5}-factor.

What would settle it

A counterexample would be a connected graph G with minimum degree δ whose size exceeds the paper's bound or whose spectral radius exceeds the threshold, yet has no spanning subgraph consisting only of P3, P4 and P5 components.

read the original abstract

Let $G$ be a connected graph of order $n$. A $\{P_3,P_4,P_5\}$-factor is a spanning subgraph $H$ of $G$ such that every component of $H$ is isomorphic to an element of $\{P_3,P_4,P_5\}$. In this paper, we establish a sufficient condition on the size of the graph $G$ with minimum degree $\delta$ to have a $\{P_3, P_4, P_5\}$-factor. Subsequently, we provide another sufficient condition on the adjacency spectral radius, ensuring that a connected graph $G$ with minimum degree $\delta$ contains a $\{P_3, P_4, P_5\}$-factor.

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 / 3 minor

Summary. The paper claims that for a connected n-vertex graph G with minimum degree δ, a lower bound on the number of edges e(G) (expressed in terms of n and δ) is sufficient to guarantee a {P3,P4,P5}-factor; likewise, a lower bound on the adjacency spectral radius ρ(G) (again in terms of n and δ) is sufficient. The proofs proceed by verifying that the stated size condition forces any potential Tutte-type obstruction for a {P3,P4,P5}-factor to vanish, after which the spectral-radius result follows from the size theorem combined with standard eigenvalue inequalities that incorporate the minimum-degree hypothesis.

Significance. If the derivations hold, the results supply new sufficient conditions for path factors of bounded length, extending the literature on factor problems by linking edge-count and spectral-radius hypotheses. The approach of deriving the spectral bound from the size bound via eigenvalue inequalities is a standard and efficient technique that respects the minimum-degree constraint.

minor comments (3)
  1. [§1] §1 (Introduction): the statement of the main size theorem should explicitly record the precise functional form of the lower bound on e(G) rather than referring only to 'a stated lower bound'.
  2. [§3] §3 (Proof of the size condition): the verification that the size hypothesis eliminates all possible obstructions would benefit from a short table or case breakdown listing the possible component types that could violate the Tutte condition.
  3. [§4] §4 (Spectral-radius result): the invocation of the eigenvalue inequality relating ρ(G) to e(G) should cite the exact reference or lemma number used, especially since the minimum-degree hypothesis is invoked to tighten the bound.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing the manuscript and recommending minor revision. The provided summary accurately captures the main results: sufficient conditions on the size and adjacency spectral radius of a connected n-vertex graph with minimum degree δ that guarantee a {P3,P4,P5}-factor. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper establishes sufficient conditions on the number of edges and on the adjacency spectral radius for a connected n-vertex graph with minimum degree δ to possess a {P3,P4,P5}-factor. These are proved by showing that the stated bounds force the non-existence of a Tutte-type obstruction for such path factors, using standard graph-theoretic lemmas on path factors and eigenvalue bounds that respect the minimum-degree hypothesis. No step reduces by construction to a fitted parameter, a self-definition, or a load-bearing self-citation whose validity depends on the present work. The central claims remain independent of the paper's own inputs and are externally falsifiable via the cited standard techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard axioms of graph theory (connectedness, minimum degree, adjacency matrix eigenvalues) and on whatever prior lemmas about path factors the authors invoke; no new entities or fitted parameters are introduced in the abstract.

axioms (2)
  • standard math Standard properties of finite simple graphs and their adjacency matrices
    Invoked throughout the statements of the theorems.
  • domain assumption Existence of path-factor theorems for shorter paths (P2, P3, etc.) that are extended here
    The paper builds on the existing literature on path factors.

pith-pipeline@v0.9.0 · 5430 in / 1243 out tokens · 25925 ms · 2026-05-09T21:09:19.301311+00:00 · methodology

discussion (0)

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Reference graph

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