Rainbow Generalizations of Ramsey Theory: A Survey

@article{Fujita2010RainbowGO,
  title={Rainbow Generalizations of Ramsey Theory: A Survey},
  author={Shinya Fujita and Colton Magnant and Kenta Ozeki},
  journal={Graphs and Combinatorics},
  year={2010},
  volume={26},
  pages={1-30}
}
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs. 
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147 Citations
Background Citations
55
Methods Citations
5
Results Citations
9

147 Citations

Citation Type

Citation Type

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References

SHOWING 1-10 OF 147 REFERENCES
Rainbow Ramsey Theory
This paper presents an overview of the current state in research directions in the rainbow Ramsey theory. We list results, problems, and conjectures related to existence of rainbow arithmetic…
Some Ramsey and anti-Ramsey results in finite groups
  • O. Serra
  • Mathematics
    Electron. Notes Discret. Math.
  • 2007
On restricted edge-colorings of bicliques
NOTE New Upper Bounds for a Canonical Ramsey Problem
TLDR
It is proved that if, then as , then as .
A note on perfect graphs
Let the lines of a complete graph be 3-colored so that no triangle gets 3 different colors. If two of these colors form perfect graphs then so does the third.
Gallai Multigraphs
A complete edge-colored graph or multigraph is called Gallai if it lacks rainbow triangles. We give a construction of all finite Gallai multigraphs.
Ramsey numbers in rainbow triangle free colorings
TLDR
This number is found precisely for a C4 and all trees on at most 6 vertices and bounds are provided for general paths.
Multicoloured Hamilton Cycles
TLDR
The edges of the complete graph $K_n$ are coloured so that no colour appears more than $\lceil cn\rceil$ times, where $c$ is the number of squares in the graph.
Edge colorings of complete graphs without tricolored triangles
We show some consequences of results of Gallai concerning edge colorings of complete graphs that contain no tricolored triangles. We prove two conjectures of Bialostocki and Voxman about the…
Note – Edge-Coloring Cliques with Three Colors on All 4-Cliques
TLDR
is constructed such that every copy of has at least three colors on its edges, and improves upon the previous probabilistic bound of due to Erdős and Gyárfás.
...
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