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Finding a monochromatic subgraph or a rainbow path

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

doi:10.1002/jgt.20179
fatcat:4vqnmieykfaulklyj5iuhnq2pu
*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. ...##
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The rainbow k-connectivity of two classes of graphs
[article]

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 . ...

##
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Constrained Ramsey Numbers
[article]

2007
*
arXiv
*
pre-print

isomorphic to S

arXiv:0706.2107v2
fatcat:5u6imyhecngt7btuldftzpqfrm
*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 . ...##
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Forbidden rainbow subgraphs that force large monochromatic or multicolored k-connected subgraphs
[article]

2019
*
arXiv
*
pre-print

Let n, k, m be positive integers with n≫ m≫ k, and let

arXiv:1811.06137v2
fatcat:uu6xohi5i5d4vmzbfymj37lzcq
*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. ...##
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Constrained Ramsey Numbers

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*. ...

##
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Constrained Ramsey Numbers

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*. ...

##
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Monochromatic and rainbow 4-connectivity of moon graphs

2019
*
IOP Conference Series: Materials Science and Engineering
*

In this paper,

doi:10.1088/1757-899x/673/1/012058
fatcat:4ky3qlsfa5hxjbkjec4gagsajm
*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*. ...##
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Colored complete hypergraphs containing no rainbow Berge triangles

2019
*
Theory and Applications of Graphs
*

The study of graph Ramsey numbers within restricted colorings, in particular forbidding

doi:10.20429/tag.2019.060201
fatcat:o7j3a45gbjcstelwt7352z7z2e
*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. ...##
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Gallai-Ramsey numbers involving a rainbow 4-path
[article]

2021
*
arXiv
*
pre-print

colored copy of G

arXiv:2109.13678v2
fatcat:4oix6rfbeba27kdfabwnhkjovq
*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. ...##
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Non-rainbow colorings of 3-, 4- and 5-connected plane graphs

2009
*
Journal of Graph Theory
*

We study vertex-colorings of plane graphs that do not contain

doi:10.1002/jgt.20414
fatcat:x7coyduqmnc6pksb4y7sgfwlbu
*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. ...##
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Rainbow Generalizations of Ramsey Theory: A Survey

2010
*
Graphs and Combinatorics
*

In this work, we collect Ramsey-type results concerning

doi:10.1007/s00373-010-0891-3
fatcat:ytvtt2g4hnhglbyx725xof43de
*rainbow*and proper edge colorings of graphs. Editions ... Similarly, in [91] , 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. ...##
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Anti-van der Waerden Numbers of Graph Products
[article]

2018
*
arXiv
*
pre-print

This result is then used to determine the anti-van der Waerden number for any Cartesian product of two

arXiv:1805.02715v1
fatcat:h3leinngw5hl5obwunfbmd4y6e
*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 . ...##
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Anti-van der Waerden Numbers of Graph Products of Cycles
[article]

2022
*
arXiv
*
pre-print

If c is

arXiv:2205.11621v1
fatcat:pfee56vpebad5feaufxlr4bzxe
*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 [13] ). ...##
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On Colorings Avoiding a Rainbow Cycle and a Fixed Monochromatic Subgraph

2010
*
Electronic Journal of Combinatorics
*

An edge coloring of

doi:10.37236/303
fatcat:r3fks7syqzhvpiqxn4rnfsjltu
*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. ...##
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Rainbow generalizations of Ramsey theory - a dynamic survey

2014
*
Theory and Applications of Graphs
*

In this work, we collect Ramsey-type results concerning

doi:10.20429/tag.2014.000101
fatcat:zawfrrlt4jc4fbtiebsbpprsli
*rainbow*edge colorings of graphs. ... Regarding*paths*in general, the following represents the best known bounds. ... Similarly, in [130] , Lefmann and Rödl considered*finding*ordered*rainbow**paths**or**monochromatic*complete graphs in edge-colored totally ordered complete graphs. ...
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