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arxiv: 1907.05546 · v1 · pith:M3PJCRFInew · submitted 2019-07-12 · 🧮 math.CO

On induced saturation for paths

Eun-Kyung Cho , Ilkyoo Choi , Boram Park This is my paper

Pith reviewed 2026-05-24 22:51 UTC · model grok-4.3

classification 🧮 math.CO
keywords induced saturationpathsP_kKneser graphsgraph theorysaturation
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The pith

P_{3n}-induced-saturated graphs exist for every positive integer n.

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

The paper proves that for every positive integer n there is a graph containing no induced copy of the path on 3n vertices, yet any addition or deletion of an edge immediately produces one. This establishes existence for all path lengths that are multiples of three and yields infinitely many distinct examples for each such length. The authors further show that every Kneser graph K(n,2) with n at least 5 is P_6-induced-saturated.

Core claim

There exists a P_{3n}-induced-saturated graph for every positive integer n. The Kneser graph K(n,2) is P_6-induced-saturated for every n≥5.

What carries the argument

Explicit constructions of graphs that contain no induced P_{3n} yet become induced-saturated under any single edge change.

If this is right

  • P_k-induced-saturated graphs exist whenever k is a multiple of 3.
  • For each fixed multiple of 3 there are infinitely many distinct examples.
  • Kneser graphs K(n,2) supply an infinite family of P_6-induced-saturated graphs.

Where Pith is reading between the lines

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

  • The same style of construction may extend to other arithmetic progressions of path lengths.
  • Induced saturation for paths appears possible for all sufficiently large k.
  • The Kneser examples suggest that strongly regular graphs could serve as witnesses for additional path lengths.

Load-bearing premise

The given constructions contain no induced copy of the target path, and every edge addition or removal creates one.

What would settle it

An explicit counterexample showing that K(5,2) contains an induced P_6, or a proof that no P_{3n}-induced-saturated graph exists for some n.

Figures

Figures reproduced from arXiv: 1907.05546 by Boram Park, Eun-Kyung Cho, Ilkyoo Choi.

Figure 1
Figure 1. Figure 1: The graph G3, and a simplified representation of the vertex set of Gn. Define the following functions from V (Gn) to V (Gn) such that for (ab, j) ∈ V (Gn), fn(ab, j) = (ab, j + 1) pn(ab, j) =    (¯ab, −j − 1) if j ≡ 0 (mod 2) (¯a¯b, −j − 1) if j ≡ 1 (mod 2) qn(ab, j) = (ac, −j + 2a), where c = a + b. In the following, if there is no confusion, we denote fn, pn, and qn by f, p, and q, respectively. It is… view at source ↗
Figure 2
Figure 2. Figure 2: The vertices surrounded by the doubled lines are in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: An induced path on 3n − 1 vertices in Gn. Claim 2.6. If there is an induced path Q of Gn in the unshaded part of the left figure of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustrations for Claim 2.6. Proof. We use the labels of the vertices in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The graph Gn − S is the disjoint union of K2 and a graph H, where H is induced by the unshaded part of [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 5
Figure 5. Figure 5: The unshaded part is isomorphic to an induced subgraph of [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: An illustration when v1 = (00, 1). When v2 = (11, 1), the shaded part does not contain a vertex in V (P) \ {v0, v1, v2}. If v3 ∈ {a1, a2, b1, b2}, then v2 ∈ {y3, y4} and so V (P) \ {v0, v1, v2} are in the unshaded part of the left figure of [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: An induced path on 3n vertices in Gn − e, where e = vw is the dashed edge. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The image of an isomorphism from G2 to H. Proof of Lemma 2.1 It is sufficient to check that each of f, p, and q moves an edge to an edge. Fix a vertex (ab, j) ∈ V (Gn) and let c = a + b ∈ Z2. See Tables 1 and 2. One can check that the set of the five vertices in the ith row of [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The image of the mapping r, where the shaded part shows the image of A under r. implies that a 0 = ¯a and −j 0 + 2a 0 = −j + 1. Suppose that a = 0. Then a 0 = 1 and j 0 = j + 1. Since (a 0 b 0 , j0 ) ∈ A and a 0 = 1, j 0 ∈ {1, 2}. Then we have j ∈ {0, 1}, which is a contradiction to (ab, j) 6∈ A. Similarly, for the case where a = 1, we have a 0 = 0 and j 0 = j − 1, which also implies that j ∈ {1, 2}, a con… view at source ↗
read the original abstract

For a graph $H$, a graph $G$ is $H$-induced-saturated if $G$ does not contain an induced copy of $H$, but either removing an edge from $G$ or adding a non-edge to $G$ creates an induced copy of $H$. Depending on the graph $H$, an $H$-induced-saturated graph does not necessarily exist. In fact, Martin and Smith (2012) showed that $P_4$-induced-saturated graphs do not exist, where $P_k$ denotes a path on $k$ vertices. Axenovich and Csik\'{o}s (2019) asked the existence of $P_k$-induced-saturated graphs for $k \ge 5$; it is easy to construct such graphs when $k\in\{2, 3\}$. Recently, R\"{a}ty constructed a graph that is $P_6$-induced-saturated. In this paper, we show that there exists a $P_{k}$-induced-saturated graph for infinitely many values of $k$. To be precise, we find a $P_{3n}$-induced-saturated graph for every positive integer $n$. As a consequence, for each positive integer $n$, we construct infinitely many $P_{3n}$-induced-saturated graphs. We also show that the Kneser graph $K(n,2)$ is $P_6$-induced-saturated for every $n\ge 5$.

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 proves existence results for induced saturation with respect to paths. It constructs, for every positive integer n, a graph G that is P_{3n}-induced-saturated (contains no induced P_{3n} but every edge addition or deletion creates one) and shows that there are in fact infinitely many such graphs for each n. Separately, it proves that the Kneser graph K(n,2) is P_6-induced-saturated for all n≥5.

Significance. The result affirmatively answers the existence question of Axenovich and Csikós for all path lengths that are multiples of 3, supplying explicit constructions rather than non-constructive arguments. The Kneser-graph family supplies an infinite concrete family of P_6-induced-saturated graphs. These are the first known constructions for any k>6 and therefore constitute a substantive contribution to the theory of induced saturation.

minor comments (3)
  1. §1, paragraph following the definition of induced saturation: the sentence 'Depending on the graph H, an H-induced-saturated graph does not necessarily exist' would benefit from a forward reference to the Martin-Smith non-existence result for P_4 so that readers immediately see the contrast with the positive results proved later.
  2. The statement that 'we construct infinitely many P_{3n}-induced-saturated graphs' for each n appears both in the abstract and in the introduction; a single consolidated statement with a precise theorem number would avoid repetition.
  3. Notation for the constructed graphs (presumably defined in §3 or §4) should be introduced once with a clear label so that later references to 'the graph G_n' are unambiguous.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending minor revision. The report correctly identifies the main contributions: explicit constructions of P_{3n}-induced-saturated graphs for every n, the infinitude result for each such n, and the infinite family of Kneser graphs K(n,2) that are P_6-induced-saturated for n≥5. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper proves existence of P_{3n}-induced-saturated graphs for every n via explicit constructions, plus a direct verification that Kneser graphs K(n,2) satisfy the P_6-induced-saturation property for n≥5. These rest on case-by-case checks that the graphs contain no induced copy of the target path but every edge addition or deletion creates one. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear; the cited prior results (Martin-Smith, Axenovich-Csikós, Ráty) are external and non-overlapping with the present authors. The derivation chain is therefore self-contained against external benchmarks and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper operates entirely within standard graph theory; no free parameters, ad-hoc axioms, or new entities are introduced.

axioms (1)
  • standard math Standard axioms and definitions of simple undirected graphs, induced subgraphs, and paths.
    Invoked throughout to define induced saturation.

pith-pipeline@v0.9.0 · 5797 in / 1264 out tokens · 31713 ms · 2026-05-24T22:51:02.913325+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · 1 internal anchor

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