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arxiv: 2605.19508 · v1 · pith:LPOFT7TYnew · submitted 2026-05-19 · 🧮 math.CO

On hamiltonian cycles of 1-tough (P₂ cup kP₁)-free graphs

Masahiro Sanka This is my paper

Pith reviewed 2026-05-20 04:18 UTC · model grok-4.3

classification 🧮 math.CO MSC 05C45
keywords hamiltonian cyclestough graphsinduced subgraph freegraph connectivityminimum degreeforbidden subgraphs
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0 comments X

The pith

1-tough and (k-1)-connected (P2 union kP1)-free graphs with minimum degree at least k are Hamiltonian when they have at least k squared plus k plus one vertices.

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

The paper aims to prove that for each integer k of four or more, a graph avoiding the induced subgraph consisting of a path on two vertices together with k isolated vertices is Hamiltonian if it is 1-tough, (k-1)-connected, has minimum degree at least k, and has at least k squared plus k plus one vertices. This matters to a reader interested in when graphs guarantee a cycle through every vertex because the conditions rule out local and global obstructions to such a cycle. The result shows that the question of Hamiltonicity for these graphs holds for all large enough examples. It combines the forbidden induced subgraph with toughness to force the desired cycle.

Core claim

For each integer k greater than or equal to 4, if G is a 1-tough and (k-1)-connected (P2 ∪ kP1)-free graph with |V(G)| at least k squared plus k plus one and minimum degree at least k, then G is Hamiltonian.

What carries the argument

The (P2 ∪ kP1)-free condition as an induced subgraph restriction that, together with 1-toughness, prevents structures incompatible with a Hamiltonian cycle under the given connectivity and size.

If this is right

  • The question of Hamiltonicity for 1-tough and k-connected (P2 ∪ kP1)-free graphs receives an affirmative answer for all graphs above the size threshold.
  • The connectivity requirement can be relaxed to (k-1) for sufficiently large graphs.
  • A minimum degree of k is enough to guarantee the Hamiltonian cycle when the other conditions hold.

Where Pith is reading between the lines

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

  • The size bound of k squared plus k plus one might be lowered with a tighter analysis of the structural restrictions.
  • The same combination of toughness and forbidden induced subgraphs could be tested for Hamiltonicity in related classes of graphs with different connectivity levels.

Load-bearing premise

The graph has at least k squared plus k plus one vertices.

What would settle it

A counterexample consisting of a 1-tough (k-1)-connected (P2 ∪ kP1)-free graph with minimum degree k and at least k squared plus k plus one vertices that does not have a Hamiltonian cycle would show the claim is false.

Figures

Figures reproduced from arXiv: 2605.19508 by Masahiro Sanka.

Figure 1
Figure 1. Figure 1: A non-hamiltonian 1-tough and 3-connected ( [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
read the original abstract

Let $k$ be a positive integer. A graph is said to be $(P_2 \cup kP_1)$-free if it does not contain $P_2 \cup kP_1$ as an induced subgraph. Recently, Ota and the author asked whether every 1-tough and $k$-connected $(P_2 \cup kP_1)$-free graph is hamiltonian or the Petersen graph. Note that this problem is affirmative for $k \in \{1,2,3\}$ by the known results. In this paper, we show that for each integer $k \geq 4$, if $G$ is a $1$-tough and $(k-1)$-connected $(P_2 \cup kP_1)$-free graph with $|V(G)| \ge k^2+k+1$ and $\delta(G) \ge k$, then $G$ is hamiltonian. This result implies that the above question is affirmative for large graphs.

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 proves that for each integer k ≥ 4, every 1-tough and (k-1)-connected (P₂ ∪ kP₁)-free graph G satisfying |V(G)| ≥ k² + k + 1 and δ(G) ≥ k is Hamiltonian. This partially affirms an open question of Ota and the author on Hamiltonicity of 1-tough k-connected (P₂ ∪ kP₁)-free graphs (or the Petersen graph) by establishing the conclusion under the stated size, degree, and connectivity hypotheses for sufficiently large graphs.

Significance. If the result holds, it contributes a concrete advance to the literature on Hamiltonian cycles in graphs with forbidden induced subgraphs and toughness conditions. The argument relies on longest-cycle techniques that exploit the (P₂ ∪ kP₁)-free hypothesis to bound neighborhoods and independent sets, together with the 1-toughness and minimum-degree assumptions to derive a contradiction from a non-Hamiltonian cycle; the explicit order threshold makes the ensuing case analysis tractable. The manuscript supplies the necessary structural lemmas and case distinctions.

minor comments (2)
  1. [Introduction] Introduction, paragraph 2: the citation to the question posed by Ota and the author should include the precise reference (year, journal or arXiv number) for immediate accessibility.
  2. [Section 3] The proof of the main theorem proceeds via a longest cycle C and analysis of the components of G−C; a short remark on why the bound |V(G)| ≥ k² + k + 1 is chosen (rather than a smaller explicit function of k) would clarify the sharpness discussion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript, accurate summary of the main theorem, and recommendation to accept. The report correctly identifies the result as a partial affirmation of the open question posed by Ota and the author, under the additional hypotheses of minimum degree at least k and order at least k² + k + 1.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via case analysis on forbidden subgraphs

full rationale

The paper states a conditional theorem: for each k ≥ 4, any 1-tough (k-1)-connected (P₂ ∪ kP₁)-free graph on at least k² + k + 1 vertices with minimum degree at least k is Hamiltonian. This is proved by standard longest-cycle arguments that use the forbidden induced subgraph to bound neighborhoods and independent sets. The size threshold is an explicit hypothesis in the statement, not a derived quantity. The prior question posed by Ota and the author is mentioned only for context and does not supply any load-bearing step or uniqueness claim inside the proof. No equations, fitted parameters, or self-referential definitions appear; the manuscript supplies direct case analysis without reducing the central claim to its own inputs or to a self-citation chain. The result is therefore independent of the listed circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard definitions of toughness, induced-subgraph freeness, connectivity, and minimum degree together with the known affirmative cases for k=1,2,3. No free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • standard math Standard definitions and basic properties of graphs, induced subgraphs, toughness, and Hamiltonicity from graph theory.
    These background notions are presupposed throughout the statement.

pith-pipeline@v0.9.0 · 5714 in / 1211 out tokens · 48805 ms · 2026-05-20T04:18:31.711976+00:00 · methodology

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Reference graph

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