Pith. Sign in

RECORD

Proper rainbow saturation for trees

PaperPDF from arXiv

data files

Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
Andrew Lane, Natasha Morrison

Combinatorics · Submitted 23 Sep 2024

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We obtain exact and asymptotic results on $\text{sat}^*(n,T)$ for several infinite families of trees.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗
Cite ▾
arXiv ID2409.15275 (version 2)
Pith IDJB46DJCP, Pith's permanent record of this paper and its review
CitationAndrew Lane, Natasha Morrison. "Proper rainbow saturation for trees." arXiv:2409.15275 (2024). https://pith.science/paper/JB46DJCP
Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Open the compiled record

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,326 input tokens.

What the abstract itself states, as probabilities: claims first 0.48 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.24 · compares with prior results 0.43 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

Given a graph $H$, we say that a graph $G$ is properly rainbow $H$-saturated if: (1) There is a proper edge colouring of $G$ containing no rainbow copy of $H$; (2) For every $e \notin E(G)$, every proper edge colouring of $G+e$ contains a rainbow copy of $H$. The proper rainbow saturation number $\text{sat}^*(n,H)$ is the minimum number of edges in a properly rainbow $H$-saturated graph. In this paper we initiate a systematic study of the proper rainbow saturation number for trees. We obtain exact and asymptotic results on $\text{sat}^*(n,T)$ for several infinite families of trees. Our proofs reveal connections to the classical saturation and semi-saturation numbers.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Given a graph $H$, we say that a graph $G$ is properly rainbow $H$-saturated if: (1) There is a proper edge colouring of $G$ containing no rainbow copy of $H$; (2) For every $e \notin E(G)$, every proper edge colouring of $G+e$ contains a rainbow copy of $H$. background 1.00 0.06 0.03 asserted 0.88
s1 The proper rainbow saturation number $\text{sat}^*(n,H)$ is the minimum number of edges in a properly rainbow $H$-saturated graph. background 0.99 0.10 0.04 asserted 0.94
s2 In this paper we initiate a systematic study of the proper rainbow saturation number for trees. contribution 0.99 0.83 0.06 asserted 0.98
s3 We obtain exact and asymptotic results on $\text{sat}^*(n,T)$ for several infinite families of trees. result 0.90 0.97 0.72 asserted 1.00
s4 Our proofs reveal connections to the classical saturation and semi-saturation numbers. interpretation 0.95 0.89 0.10 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Given a graph $H$, we say that a graph $G$ is properly rainbow $H$-saturated if: (1) There is a proper edge colouring of $G$ containing no rainbow copy of $H$; (2) For every $e \notin E(G)$, every proper edge colouring of $G+e$ contains a rainbow copy of $H$. The proper rainbow saturation number $\text{sat}^*(n,H)$ is the minimum number of edges in a properly rainbow $H$-saturated graph. In this paper we initiate a systematic study of the proper rainbow saturation number for trees. We obtain exact and asymptotic results on $\text{sat}^*(n,T)$ for several infinite families of trees. Our proofs reveal connections to the classical saturation and semi-saturation numbers.

Human review

How Pith checks papers → · Full audit of this paper →

References
How the literature uses it

What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites

Compares with 1 paper
  • Prior to this work, asymptotically tight bounds on $(n,H)$ were only known for $H = C_4$ (Halfpap, Lidický, and Masařík [ref] ), and $H = P_4$ (Bushaw, Johnston, and Rombach [ref] ).

    2403.15602 · Introduction · 0.88

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.