RECORD
Rainbow cycles through specified vertices
Combinatorics · Submitted 30 May 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim One of the results is a classification of all graphs $G$ such that $crx_k(G)=e(G)$, for $k=1,2$.
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.
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The abstract, sentence by sentence
An edge-coloured cycle is rainbow if the edges have distinct colours. Let $G$ be a graph such that any $k$ vertices lie in a cycle of $G$. The $k$-rainbow cycle index of $G$, denoted by $crx_k(G)$, is the minimum number of colours required to colour the edges of $G$ such that, for every set $S$ of $k$ vertices in $G$, there exists a rainbow cycle in $G$ containing $S$. In this paper, we will first prove some results about the parameter $crx_k(G)$ for general graphs $G$. One of the results is a classification of all graphs $G$ such that $crx_k(G)=e(G)$, for $k=1,2$. We will also determine $crx_k(G)$ for some specific graphs $G$, including wheels, complete graphs, complete bipartite and multipartite graphs, and discrete cubes.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | An edge-coloured cycle is rainbow if the edges have distinct colours. | background 1.00 | 0.05 | 0.02 | asserted 1.00 |
| s1 | Let $G$ be a graph such that any $k$ vertices lie in a cycle of $G$. | background 0.99 | 0.05 | 0.02 | asserted 0.85 |
| s2 | The $k$-rainbow cycle index of $G$, denoted by $crx_k(G)$, is the minimum number of colours required to colour the edges of $G$ such that, for every set $S$ of $k$ vertices in $G$, there exists a rainbow cycle in $G$ containing $S$. | background 0.93 | 0.11 | 0.06 | asserted 0.98 |
| s3 | In this paper, we will first prove some results about the parameter $crx_k(G)$ for general graphs $G$. | contribution 0.96 | 0.73 | 0.04 | asserted 0.94 |
| s4 | One of the results is a classification of all graphs $G$ such that $crx_k(G)=e(G)$, for $k=1,2$. | result 0.77 | 0.91 | 0.19 | asserted 1.00 |
| s5 | We will also determine $crx_k(G)$ for some specific graphs $G$, including wheels, complete graphs, complete bipartite and multipartite graphs, and discrete cubes. | contribution 0.75 | 0.93 | 0.17 | asserted 1.00 |
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The paper
Original abstract
An edge-coloured cycle is rainbow if the edges have distinct colours. Let $G$ be a graph such that any $k$ vertices lie in a cycle of $G$. The $k$-rainbow cycle index of $G$, denoted by $crx_k(G)$, is the minimum number of colours required to colour the edges of $G$ such that, for every set $S$ of $k$ vertices in $G$, there exists a rainbow cycle in $G$ containing $S$. In this paper, we will first prove some results about the parameter $crx_k(G)$ for general graphs $G$. One of the results is a classification of all graphs $G$ such that $crx_k(G)=e(G)$, for $k=1,2$. We will also determine $crx_k(G)$ for some specific graphs $G$, including wheels, complete graphs, complete bipartite and multipartite graphs, and discrete cubes.
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