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Rainbow cycles in flip graphs
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Felsner, Stefan, Kleist, Linda, Mutze, Torsten and Sering, Leon (2020) Rainbow cycles in flip graphs. SIAM Journal on Discrete Mathematics, 34 (1). pp. 1-39. doi:10.1137/18M1216456
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Official URL: https://doi.org/10.1137/18M1216456
Abstract
The flip graph of triangulations has as vertices all triangulations of a convex $n$-gon, and an edge between any two triangulations that differ in exactly one edge. An $r$-rainbow cycle in this graph is a cycle in which every inner edge of the triangulation appears exactly $r$~times. This notion of a rainbow cycle extends in a natural way to other flip graphs. In this paper we investigate the existence of $r$-rainbow cycles for three different flip graphs on classes of geometric objects: the aforementioned flip graph of triangulations of a convex $n$-gon, the flip graph of plane trees on an arbitrary set of $n$~points, and the flip graph of non-crossing perfect matchings on a set of $n$~points in convex position. In addition, we consider two flip graphs on classes of non-geometric objects: the flip graph of permutations of $\{1,2,\dots,n\}$ and the flip graph of $k$-element subsets of $\{1,2,\dots,n\}$. In each of the five settings, we prove the existence and non-existence of rainbow cycles for different values of~$r$, $n$ and~$k$.
| Item Type: | Journal Article | ||||||
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| Alternative Title: | |||||||
| Subjects: | Q Science > QA Mathematics Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software |
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| Divisions: | Faculty of Science > Computer Science | ||||||
| Library of Congress Subject Headings (LCSH): | Graph theory, Gray codes, Spanning trees (Graph theory), Computer science -- Mathematics | ||||||
| Journal or Publication Title: | SIAM Journal on Discrete Mathematics | ||||||
| Publisher: | Society for Industrial and Applied Mathematics | ||||||
| ISSN: | 0895-4801 | ||||||
| Official Date: | 2 January 2020 | ||||||
| Dates: |
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| Volume: | 34 | ||||||
| Number: | 1 | ||||||
| Page Range: | pp. 1-39 | ||||||
| DOI: | 10.1137/18M1216456 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Publisher Statement: | “First Published in SIAM Journal on Discrete Mathematics in 34,1, 2020, published by the Society for Industrial and Applied Mathematics (SIAM)” and the copyright notice as stated in the article itself (e.g., “Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.”) | ||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||
| Copyright Holders: | © 2020, Society for Industrial and Applied Mathematics | ||||||
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| Open Access Version: |
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