Rainbow faces in edge‐colored plane graphs

Jendroľ, Stanislav; Miškuf, Jozef; Soták, Roman; Škrabuľáková, Erika

doi:10.1002/jgt.20382

Cited by 10 publications

(5 citation statements)

References 25 publications

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“…Because such nodes are often minor, some links can be deleted in A-6). In A-5), a link between two nodes with a larger distance should be deleted to support the design of a simple plane graph, which will be useful to other research areas [49]. The computational complexity of this algorithm is presented in the Appendix.…”

Section: A-4)mentioning

confidence: 99%

Metropolitan Area Network Model Design Using Regional Railways Information for Beyond 5G Research

Tachibana

Hirota²,

SUZUKI

et al. 2023

IEICE Trans. Commun.

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To accelerate research on Beyond 5G (B5G) technologies in Japan, we propose an algorithm that designs mesh-type metropolitan area network (MAN) models based on a priori Japanese regional railway information, because ground-truth communication network information is unavailable. Instead, we use the information of regional railways, which is expected to express the necessary geometric structure of our metropolitan cities while remaining strongly correlated with their population densities and demographic variations. We provide an additional compression algorithm for use in reducing a small-scale network model from the original MAN model designed using the proposed algorithm. Two Tokyo MAN models are created, and we provide day and night variants for each while highlighting the number of passengers alighting/boarding at each station and the respective population densities. The validity of the proposed algorithm is verified through comparisons with the Japan Photonic Network model and another model designed using the communication network information, which is not ground-truth. Comparison results show that our proposed algorithm is effective for designing MAN models and that our result provides a valid Tokyo MAN model.

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Section: A-4)mentioning

confidence: 99%

Metropolitan Area Network Model Design Using Regional Railways Information for Beyond 5G Research

Tachibana

Hirota²,

SUZUKI

et al. 2023

IEICE Trans. Commun.

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“…We obtain the following analogous result. Colorings of plane graphs that avoid rainbow faces have also been studied, see, e.g., [5,7,15,16]. Various results on anti-Ramsey numbers can be found in: [1,2,8,9,10,11,13,14] to name a few.…”

Section: Introductionmentioning

confidence: 99%

Planar anti-Ramsey numbers of paths and cycles

Lan

Shi

Song

2019

Discrete Mathematics

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Motivated by anti-Ramsey numbers introduced by Erdős, Simonovits and Sós in 1975, we study the anti-Ramsey problem when host graphs are plane triangulations. Given a positive integer n and a planar graph H, let T n (H) be the family of all plane triangulations T on n vertices such that T contains a subgraph isomorphic to H. The planar anti-Ramsey number of H, denoted ar P (n, H), is the maximum number of colors in an edge-coloring of a plane triangulation T ∈ T n (H) such that T contains no rainbow copy of H. Analogous to anti-Ramsey numbers and Turán numbers, planar anti-Ramsey numbers are closely related to planar Turán numbers, where the planar Turán number of H is the maximum number of edges of a planar graph on n vertices without containing H as a subgraph. The study of ar P (n, H) (under the name of rainbow numbers) was initiated by Horňák, Jendrol ′ , Schiermeyer and Soták [J Graph Theory 78 (2015) 248-257]. In this paper we study planar anti-Ramsey numbers for paths and cycles. We first establish lower bounds for ar P (n, P k ) when n ≥ k ≥ 8. We then improve the existing lower bound for ar P (n, C k ) when k ≥ 5 and n ≥ k 2 − k. Finally, using the main ideas in the above-mentioned paper, we obtain upper bounds for ar P (n, C 6 ) when n ≥ 8 and ar P (n, C 7 ) when n ≥ 13, respectively.

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“…Results on planar anti-Ramsey numbers of paths and cycles can be found in [6,13]. Colorings of plane graphs that avoid rainbow faces have also been studied, see, e.g., [5,7,17,18]. Various results on anti-Ramsey numbers can be found in: [1,2,9,10,11,14,16] to name a few.…”

Section: Introductionmentioning

confidence: 99%

Planar anti-Ramsey numbers of matchings

Chen

Lan

Song

2019

Discrete Mathematics

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Given a positive integer n and a planar graph H, let T n (H) be the family of all plane triangulations T on n vertices such that T contains a subgraph isomorphic to H. The planar anti-Ramsey number of H, denoted ar P (n, H), is the maximum number of colors in an edge-coloring of a plane triangulation T ∈ T n (H) such that T contains no rainbow copy of H. In this paper we study planar anti-Ramsey numbers of matchings. For all t ≥ 1, let M t denote a matching of size t. We prove that for all t ≥ 6 and n ≥ 3t − 6, 2n + 3t − 15 ≤ ar P (n, M t ) ≤ 2n + 4t − 14, which significantly improves the existing lower and upper bounds for ar P (n, M t ). It seems that for each t ≥ 6, the lower bound we obtained is the exact value of ar P (n, M t ) for sufficiently large n. This is indeed the case for M 6 . We prove that ar P (n, M 6 ) = 2n + 3 for all n ≥ 30.

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Rainbow faces in edge‐colored plane graphs

Cited by 10 publications

References 25 publications

Metropolitan Area Network Model Design Using Regional Railways Information for Beyond 5G Research

Metropolitan Area Network Model Design Using Regional Railways Information for Beyond 5G Research

Planar anti-Ramsey numbers of paths and cycles

Planar anti-Ramsey numbers of matchings

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