Ramsey and Gallai-Ramsey Numbers for Two Classes of Unicyclic Graphs

Wang, Zhao; Mao, Yaping; Magnant, Colton; Zou, Jinyu

doi:10.1007/s00373-020-02248-8

Cited by 8 publications

(4 citation statements)

References 13 publications

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“…With the additional restriction of forbidding the rainbow copy of G, it is clear that gr k (G : H) ≤ R k (H) for any G. Till now, most work focuses on the case G = K 3 ; see [3,6,8,11,11,15,16,20,21,22,30]. For more details on the Gallai-Ramsey numbers, we refer to the book [19] and a survey paper [7].…”

Section: Gallai-ramsey Numbermentioning

confidence: 99%

Gallai-Ramsey numbers involving a rainbow $4$-path

Zou¹,

Wang²,

Lai³

et al. 2021

Preprint

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Given two non-empty graphs G, H and a positive integer k, the Gallai-Ramsey number gr k (G : H) is defined as the minimum integer N such that for all n ≥ N , every k-edge-coloring of K n contains either a rainbow colored copy of G or a monochromatic copy of H. In this paper, we got some exact values or bounds for gr k (P 5 : H) (k ≥ 3) if H is a general graph or a star with extra independent edges or a pineapple.

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Section: Gallai-ramsey Numbermentioning

confidence: 99%

Gallai-Ramsey numbers involving a rainbow $4$-path

Zou¹,

Wang²,

Lai³

et al. 2021

Preprint

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“…More results about Gallai-edge-colorings and Gallai-Ramsey numbers can be found in [6,7,11,12,13,15,17,19,22,23,24,26,27]. And a survey can be found in [8].…”

Section: Introductionmentioning

confidence: 99%

Complete Bipartite Graphs Without Small Rainbow Stars

Chen

Mao

et al. 2023

Preprint

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“…The cases for t ≥ 5 are still open. There are also a lot of results on Gallai-Ramsey numbers for other monochromatic subgraphs, such as bipartite graphs [34,122,175], cycles and paths [23,26,34,78,93,116,164,172,177,178], stars [91,119], fans [138], wheels [163], double stars [106], books [181] and other specific graphs [139,173,179,180]. Gallai-Ramsey numbers have also been generalized to hypergraphs; see [27].…”

Section: Generalizations Of Ramsey Theorymentioning

confidence: 99%

“…. , F 8 have been determined by Faudree et al [72] for F 1 and F 2 , Gyárfás et al [91] for F 3 , Wang et al [173] for F 4 and F 5 , Fujita and Magnant [78] for F 6 , Wu et al [175] for F 7 , and Mao et al [138] for F 8 . In the subsequent sections, we will determine the Gallai-Ramsey numbers for all the remaining graphs in this class.…”

Section: Introductionmentioning

confidence: 99%

Gallai-Ramsey numbers for graphs and their generalizations

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This thesis contains research results on Gallai-Ramsey theory and its generalizations, which were obtained by the author with collaborators between September 2017 and February 2021. Apart from an introductory chapter (Chapter 1), the reader will find six closely related technical chapters (Chapters 2-7), which are mainly based on the research results that the author obtained when he was working as a PhD student at Northwestern Polytechnical University, Xi'an and the University of Twente.Chapters 2 and 3 focus on determining the exact values of Gallai-Ramsey numbers for several graphs. The other chapters mainly focus on studying various generalizations or variants of Gallai-Ramsey theory. In Chapter 4, we consider two extremal problems related to Gallai-colorings. In Chapter 5, we study the Erdős-Gyárfás function with respect to Gallai-colorings. In Chapter 6, we present a forbidden rainbow subgraph condition for an edge-colored graph to have a highly-connected monochromatic subgraph. In Chapter 7, we deal with the rainbow Erdős-Rothschild problem with respect to 3-term arithmetic progressions. Papers underlying this thesis[1] Gallai-Ramsey numbers for a class of graphs with five vertices, Graphs and Combinatorics 36 (2020), 1603-1618 (with L. Wang). (Chapter 2) [2] Extremal problems and results related to Gallai-colorings, Discrete Mathematics 344 (2021), 112567 (with H.J. Broersma and L. Wang). (Chapters 3 and 4) vii viii Preface [3] The Erdős-Gyárfás function with respect to Gallai-colorings, submitted (with H.J. Broersma and L. Wang). (Chapter 5) [4] Forbidden rainbow subgraphs that force large monochromatic or multicolored k-connected subgraphs, Discrete Applied Mathematics 285 (2020), 18-27 (with L. Wang).(Chapter 6)[5] Integer colorings with no rainbow 3-term arithmetic progression, submitted (with H.J. Broersma and L. Wang).

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Ramsey and Gallai-Ramsey Numbers for Two Classes of Unicyclic Graphs

Cited by 8 publications

References 13 publications

Gallai-Ramsey numbers involving a rainbow $4$-path

Gallai-Ramsey numbers involving a rainbow $4$-path

Complete Bipartite Graphs Without Small Rainbow Stars

Gallai-Ramsey numbers for graphs and their generalizations

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