Edges not in any monochromatic copy of a fixed graph

Liu, Hong; Pikhurko, Oleg; Sharifzadeh, Maryam

doi:10.1016/j.jctb.2018.07.007

Cited by 10 publications

(17 citation statements)

References 41 publications

(70 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Ramsey-Turán number with more than 2 colours. We remark that the multicolour Ramsey-Turán number for triangles is related to a version of Ramsey number studied by Liu, Pikhurko and Sharifzadeh [19]. They introduced r * (K a 1 , .…”

Section: 3mentioning

confidence: 90%

See 1 more Smart Citation

Two Conjectures in Ramsey--Turán Theory

Kim¹,

Kim²,

Liu³

2019

SIAM J. Discrete Math.

Self Cite

View full text Add to dashboard Cite

show abstract

Section: 3mentioning

confidence: 90%

“…In particular, Theorems 1.8 and 1.9 in [19] imply ̺(K 3 , K 3 , K 3 ) = 2 5 and ̺(K 3 , K 3 , K 3 , K 3 ) = 15 32 . In general Ramsey-Turán numbers for larger cliques are not determined by r * , for example ̺(K 3 , K 5 ) = 2 5 = 1 2 1 − 1 r * (K 3 ,K 5 ) = 3 8 and ̺(K 4 , K 4 ) = 11 28 = 1 2 1 − 1 r * (K 4 ,K 4 ) = 1 3 .…”

Section: 3mentioning

confidence: 97%

Two Conjectures in Ramsey--Turán Theory

Kim¹,

Kim²,

Liu³

2019

SIAM J. Discrete Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…In 2017, Ma [14] confirmed their problem for an infinite family of bipartite graphs. Later, Liu et al [13] extended Ma's result to a larger family of bipartite graphs and proved an upper bound for all bipartite graphs.…”

Section: Introductionmentioning

confidence: 99%

Extremal graphs for odd wheels

Yuan

2021

Journal of Graph Theory

View full text Add to dashboard Cite

For a graph H, the Turán number of H, denoted by ex(n,H), is the maximum number of edges of an n‐vertex H‐free graph. Let g(n,H) denote the maximum number of edges not contained in any monochromatic copy of H in a 2‐edge‐coloring of Kn. A wheel Wm is a graph formed by connecting a single vertex to all vertices of a cycle of length m−1. The Turán number of W2k was determined by Simonovits in 1960s. In this paper, we determine ex(n,W2k+1) when n is sufficiently large. We also show that, for sufficient large n, g(n,W2k+1)=ex(n,W2k+1) which confirms a conjecture posed by Keevash and Sudakov for odd wheels.

show abstract

“…Note that f k (n, K 3 ) ≤ nim k (n, K 3 ). For sufficiently large n, since nim 2 (n, K 3 ) = t(n, 2) (proven in [107]) and nim 3 (n, K 3 ) = t(n, 5) (proven in [128]), we have f k (n, K 3 ) = t(n, g r k−1 (K 3 : K 3 ) − 1) for k ∈ {2, 3}. In the following, we may assume k ≥ 4.…”

Section: Note That We Have Grmentioning

confidence: 91%

“…Recall that the Gallai-Ramsey number g r k (K 3 : H) is the minimum integer n such that every k-edge-coloring of K n contains either a rainbow copy of K 3 or a monochromatic copy of H. As we have seen in the previous two chapters, it is far from trivial to determine exact values of Gallai-Ramsey numbers, even for a small graph H. There the lower bounds were obtained by showing specific k-edge-colorings of complete graphs that do not admit a rainbow copy of K 3 or a monochromatic copy of H. Instead of disallowing a rainbow copy of K 3 and a monochromatic copy of H, it is natural to consider the related problem of determining the maximum number of edges that are not contained in any rainbow copy of K 3 or monochromatic copy of H. The analogous problem for classical Ramsey numbers was considered in [107,128,133]; in these papers the authors studied the maximum number of edges not contained in any monochromatic copy of H over all k-edge-colorings of K n . For k ≥ 2, let f k (n, H) denote the maximum number of edges not contained in any rainbow triangle or monochromatic copy of H, over all k-edge-colorings of K n .…”

Section: Introductionmentioning

confidence: 99%

Gallai-Ramsey numbers for graphs and their generalizations

View full text Add to dashboard Cite

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).

show abstract

Edges not in any monochromatic copy of a fixed graph

Abstract: For a sequence (H i

Cited by 10 publications

References 41 publications

Two Conjectures in Ramsey--Turán Theory

Two Conjectures in Ramsey--Turán Theory

Extremal graphs for odd wheels

Gallai-Ramsey numbers for graphs and their generalizations

Contact Info

Product

Resources

About