Rainbow generalizations of Ramsey theory - a dynamic survey

Fujita, Shinya; Magnant, Colton; Ozeki, Kenta

doi:10.20429/tag.2014.000101

Cited by 68 publications

(59 citation statements)

References 132 publications

(159 reference statements)

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“…Chen and Daykin [5] showed that for each integers ℓ t n , , with ≥ ≥ t n t 2, 25 , and ≥ ℓ ≥ n 2, an edge-coloring of the complete bipartite graph K n n , admits a PC ℓ C 2 if no vertex is incident to t edges of the same color. More related results can be found in surveys [8,9].…”

Section: Color Degree Conditionmentioning

confidence: 58%

“…Chen and Daykin showed that for each integers

t goodbreakinfix, n goodbreakinfix, ℓ

with

t goodbreakinfix\geq 2 goodbreakinfix, n goodbreakinfix\geq 25 t

, and

n goodbreakinfix\geq ℓ goodbreakinfix\geq 2

, an edge‐coloring of the complete bipartite graph

K_{n, n}

admits a PC

C_{2 ℓ}

if no vertex is incident to

t

edges of the same color. More related results can be found in surveys .…”

Section: Introductionmentioning

confidence: 69%

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A complete bipartite graph without properly colored cycles of length four

Čada

Ozeki

Yoshimoto

2019

Journal of Graph Theory

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A subgraph of an edge‐colored graph is said to be properly colored, or shortly PC, if any two adjacent edges have different colors. Fujita, Li, and Zhang gave a decomposition theorem for edge‐colorings of complete bipartite graphs without PC C4. However, their decomposition just focuses on a local structure. In this paper, we give a new and global decomposition theorem for edge‐colorings of complete bipartite graphs without PC C4. Our decomposition gives a corollary on the existence of a monochromatic star with almost sharp bound.

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Section: Color Degree Conditionmentioning

confidence: 58%

“…Chen and Daykin showed that for each integers

t goodbreakinfix, n goodbreakinfix, ℓ

with

t goodbreakinfix\geq 2 goodbreakinfix, n goodbreakinfix\geq 25 t

, and

n goodbreakinfix\geq ℓ goodbreakinfix\geq 2

, an edge‐coloring of the complete bipartite graph

K_{n, n}

admits a PC

C_{2 ℓ}

if no vertex is incident to

t

edges of the same color. More related results can be found in surveys .…”

Section: Introductionmentioning

confidence: 69%

A complete bipartite graph without properly colored cycles of length four

Čada

Ozeki

Yoshimoto

2019

Journal of Graph Theory

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“…Other related results can be found in [2,4,11,12]. We refer the interested reader to [5] for a survey of Gallai-Ramsey numbers, with a dynamic version available at [6]. We also state the corresponding conjecture for bipartite graphs.…”

Section: Conjecture 2 ([3]mentioning

confidence: 95%

A General Lower Bound on Gallai-Ramsey Numbers for Non-Bipartite Graphs

Magnant¹

2018

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Given a graph H and a positive integer k, the k-color Gallai-Ramsey number gr k (K 3 : H) is defined to be the minimum number of vertices n for which any k-coloring of the complete graph K n contains either a rainbow triangle or a monochromatic copy of H. The behavior of these numbers is rather well understood when H is bipartite but when H is not bipartite, this behavior is a bit more complicated. In this short note, we improve upon existing lower bounds for non-bipartite graphs H to a value that we conjecture to be sharp up to a constant multiple.

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“…We refer to the survey of rainbow generalizations of Ramsey Theory [10,11] for more information on this topic and a complete list of known results involving Gallai-Ramsey numbers. We are now able to state the conjecture of Fox et al…”

Section: Introductionmentioning

confidence: 99%

All partitions have small parts — Gallai–Ramsey numbers of bipartite graphs

Magnant

Nowbandegani

et al. 2019

Discrete Applied Mathematics

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Gallai-colorings are edge-colored complete graphs in which there are no rainbow triangles. Within such colored complete graphs, we consider Ramsey-type questions, looking for specified monochromatic graphs. In this work, we consider monochromatic bipartite graphs since the numbers are known to grow more slowly than for non-bipartite graphs. The main result shows that it suffices to consider only 3colorings which have a special partition of the vertices. Using this tool, we find several sharp numbers and conjecture the sharp value for all bipartite graphs. In particular, we determine the Gallai-Ramsey numbers for all bipartite graphs with two vertices in one part and initiate the study of linear forests.

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Rainbow generalizations of Ramsey theory - a dynamic survey

Abstract: In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs.If you have corrections, updates or new results which fit the scope of this work, please contact Colton Magnant at coltonmagnant@yahoo.com.

Cited by 68 publications

References 132 publications

A complete bipartite graph without properly colored cycles of length four

A complete bipartite graph without properly colored cycles of length four

A General Lower Bound on Gallai-Ramsey Numbers for Non-Bipartite Graphs

All partitions have small parts — Gallai–Ramsey numbers of bipartite graphs

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