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### Finding a monochromatic subgraph or a rainbow path

András Gyárfás, Jenő Lehel, Richard H. Schelp
2006 Journal of Graph Theory
For simple graphs G and H, let f (G, H) denote the least integer N such that every coloring of the edges of KN contains either a monochromatic copy of G or a rainbow copy of H.  ...  Then d = 4 because (z, y, x 3 , x 0 , x i ) is a rainbow path for i = 1 or 2.  ...  Note that in this case the color of zw cannot be 1 or 3, because of the emerging rainbow paths (z, w, x 2 , x 0 , x 3 ) or (z, w, x 2 , x 0 , x 1 ), respectively.  ...

### The rainbow k-connectivity of two classes of graphs [article]

Xueliang Li, Yuefang Sun
2009 arXiv   pre-print
A path in an edge-colored graph G, where adjacent edges may be colored the same, is called a rainbow path if no two edges of G are colored the same.  ...  vertices of G are connected by k internally disjoint rainbow paths.  ...  In subgraph G 1 , we find the monochromatic u − u (i 0 −1)ℓ+t 0 ,1 path u, y, u (i 0 −1)ℓ+t 0 ,1 . Let y = u (i−1)ℓ+t,1 where i = 1, i 0 , t = 1, t 0 or y = u t 0 ,1 or u (i 0 −1)ℓ+1,1 .  ...

### Constrained Ramsey Numbers [article]

Po-Shen Loh, Benny Sudakov
2007 arXiv   pre-print
isomorphic to S or a rainbow (all edges differently colored) subgraph isomorphic to T.  ...  For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge coloring of the complete graph on n vertices, with any number of colors, has a monochromatic subgraph  ...  monochromatic subgraph isomorphic to S or a rainbow subgraph isomorphic to T .  ...

### Forbidden rainbow subgraphs that force large monochromatic or multicolored k-connected subgraphs [article]

Xihe Li, Ligong Wang
2019 arXiv   pre-print
Let n, k, m be positive integers with n≫ m≫ k, and let A be the set of graphs G of order at least 3 such that there is a k-connected monochromatic subgraph of order at least n-f(G,k,m) in any rainbow G-free  ...  Moreover, we show that for any graph H∈A, if n sufficiently larger than m and k, then any rainbow (P_3∪ H)-free coloring of K_n using all the m colors contains a k-connected monochromatic subgraph of order  ...  Instead to find a connected monochromatic subgraph, it will also be interesting to looking for a long monochromatic paths or cycles in colored complete graphs.  ...

### Constrained Ramsey Numbers

PO-SHEN LOH, BENNY SUDAKOV
2007 Combinatorics, probability & computing
monochromatic subgraph isomorphic to S or a rainbow subgraph isomorphic to T .  ...  Both proofs find a structured subgraph G ⊂ G in which one may direct some edges in such a way that directed paths correspond to rainbow paths.  ...

### Constrained Ramsey Numbers

Po-Shen Loh, Benny Sudakov
2007 Electronic Notes in Discrete Mathematics
monochromatic subgraph isomorphic to S or a rainbow subgraph isomorphic to T .  ...  Both proofs find a structured subgraph G ⊂ G in which one may direct some edges in such a way that directed paths correspond to rainbow paths.  ...

### Monochromatic and rainbow 4-connectivity of moon graphs

M A Shulhany, D K N Rachim, Y Rukmayadi
2019 IOP Conference Series: Materials Science and Engineering
In this paper, a security system is represented by a special mathematics object, namely a graph. The study focuses on the moon graphs or Mon.  ...  Monochromatic and rainbow connectivity are known as the concepts that can calculate the minimum number of passwords and security systems.  ...  monochromatic path or 4 internally disjoint rainbow paths.  ...

### Colored complete hypergraphs containing no rainbow Berge triangles

Colton Magnant
2019 Theory and Applications of Graphs
The study of graph Ramsey numbers within restricted colorings, in particular forbidding a rainbow triangle, has recently been blossoming under the name Gallai-Ramsey numbers.  ...  In this work, we extend the main structural tool from rainbow triangle free colorings of complete graphs to rainbow Berge triangle free colorings of hypergraphs.  ...  The author would also like to thank the referees for their careful reading and constructive comments, leading to a much improved presentation of this work.  ...

### Gallai-Ramsey numbers involving a rainbow 4-path [article]

Jinyu Zou, Zhao Wang, Hong-Jian Lai, Yaping Mao
2021 arXiv   pre-print
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.  ...  rainbow copy of G or a monochromatic copy of H.  ...

### Non-rainbow colorings of 3-, 4- and 5-connected plane graphs

Zdeněk Dvořák, Daniel Král', Riste Škrekovski
2009 Journal of Graph Theory
We study vertex-colorings of plane graphs that do not contain a rainbow face, i.e., a face with vertices of mutually distinct colors.  ...  If G is a 3-connected plane graph with n vertices, then the number of colors in such a coloring does not exceed 7n−8 9 .  ...  A color class of a vertex-coloring is the set of vertices assigned the same color. A monochromatic path or cycle is a path or a cycle such that all its vertices have the same color.  ...

### Rainbow Generalizations of Ramsey Theory: A Survey

Shinya Fujita, Colton Magnant, Kenta Ozeki
2010 Graphs and Combinatorics
In this work, we collect Ramsey-type results concerning rainbow and proper edge colorings of graphs. Editions  ...  Similarly, in  , Lefmann and Rödl considered finding ordered rainbow paths or monochromatic complete graphs in edge-colored totally ordered complete graphs.  ...  rainbow copy of G or a monochromatic copy of H.  ...

### Anti-van der Waerden Numbers of Graph Products [article]

Hunter Rehm, Alex Schulte, Nathan Warnberg
2018 arXiv   pre-print
This result is then used to determine the anti-van der Waerden number for any Cartesian product of two paths.  ...  In this paper, anti-van der Waerden numbers on Cartesian products of graphs are investigated and a conjecture made by Schulte, et al (see arXiv:1802.01509) is answered.  ...  Let v k,α be the vertex in G k that corresponds to v i,α and find a shortest path P from v j,β to v k,α whose only vertex in G j is v j,β .  ...

### Anti-van der Waerden Numbers of Graph Products of Cycles [article]

Joe Miller, Nathan Warnberg
2022 arXiv   pre-print
If c is a coloring function of the vertices of G and a k-AP in G has each vertex colored distinctly, then that k-AP is a rainbow k-AP.  ...  ,r} = [r] is guaranteed to have a rainbow k-AP. This paper focuses on 3-APs and graph products with cycles.  ...  you colors the edges of a complete graph you can always find a complete subgraph that is monochromatic (see  ).  ...

### On Colorings Avoiding a Rainbow Cycle and a Fixed Monochromatic Subgraph

Maria Axenovich, JiHyeok Choi
2010 Electronic Journal of Combinatorics
An edge coloring of a complete graph is called \$(H,G)\$-good if there is no monochromatic copy of \$G\$ and no rainbow (totally multicolored) copy of \$H\$ in this coloring.  ...  As shown by Jamison and West, an \$(H,G)\$-good coloring of an arbitrarily large complete graph exists unless either \$G\$ is a star or \$H\$ is a forest.  ...  Acknowledgments The authors thank the referee for a very careful reading and useful comments improving the presentation of the results.  ...

### Rainbow generalizations of Ramsey theory - a dynamic survey

Shinya Fujita, Colton Magnant, Kenta Ozeki
2014 Theory and Applications of Graphs
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs.  ...  Regarding paths in general, the following represents the best known bounds.  ...  Similarly, in  , Lefmann and Rödl considered finding ordered rainbow paths or monochromatic complete graphs in edge-colored totally ordered complete graphs.  ...
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