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An anti-Ramsey Theorem on edge-cutsets

Juan José Montellano-Ballesteros
2006 Discussiones Mathematicae Graph Theory  
Acknowledgement I like to thank the referee for suggesting this shorter and clearer alternative proof of the theorem.  ...  Problems concerning TMC subgraphs in edge-colourings of a host graph are called anti-Ramsey problems (see [1, 2, 3, 4, 5, 6, 7] ).  ...  Theorem 1. Let G = (V (G), E(G)) be Given a (k(G) + 1)-edge-colouring of G, let H be a TMC subgraph of G of size k(G) + 1.  ... 
doi:10.7151/dmgt.1297 fatcat:27igi25hq5ctdboenl46nlqlqu

Anti-Ramsey number of matchings in r-partite r-uniform hypergraphs [article]

Yisai Xue, Erfang Shan, Liying Kang
2021 arXiv   pre-print
In this paper, as a generalization of these results, we determine the anti-Ramsey number ar_r(𝒦_n_1,...  ...  An edge-colored hypergraph is rainbow if all of its edges have different colors.  ...  An anti-Ramsey theorem on cycles. Graphs and Combin. 21(3):343–354, 2005. 12 [13] L. Özkahya and M. Young.  ... 
arXiv:2109.05163v2 fatcat:ydudpo4cdffi7ls3hn45hd7jaq

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  ...  Anti-Ramsey Theory The anti-Ramsey problem is stated as follows.  ...  In other work, Gorgol and Lazuka computed the following anti-Ramsey numbers for stars with an added edge.  ... 
doi:10.1007/s00373-010-0891-3 fatcat:ytvtt2g4hnhglbyx725xof43de

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.  ...  The authors of [68] also found the Gallai Ramsey numbers for all trees of order at most 6. Regarding paths in general, the following represents the best known bounds.  ...  Anti-Ramsey Theory The anti-Ramsey problem is stated as follows.  ... 
doi:10.20429/tag.2014.000101 fatcat:zawfrrlt4jc4fbtiebsbpprsli

Anti-Ramsey Hypergraph Numbers

Mark Budden, William Stiles
2021 Electronic Journal of Graph Theory and Applications  
The anti-Ramsey number ar n (H) of an r-uniform hypergraph is the maximum number of colors that can be used to color the hyperedges of a complete r-uniform hypergraph on n vertices without producing a  ...  In this paper, we determine anti-Ramsey numbers for paths of length 2, certain stars and complete hypergraphs, and the complete 3-uniform hypergraph of order 4 with a single hyperedge removed.  ...  Let c : E(K Applying Theorem 2.2 to known anti-Ramsey numbers on graphs, we can obtain select results on hypergraphs.  ... 
doi:10.5614/ejgta.2021.9.2.12 fatcat:6znwypdkcrf2digyjpge4agftq

Low Rank Co-Diagonal Matrices and Ramsey Graphs

Vince Grolmusz
2000 Electronic Journal of Combinatorics  
We also show, that explicit constructions of such low rank matrices imply explicit constructions of Ramsey graphs.  ...  The author is grateful to Noga Alon for his comments and for his significant improvement of Theorem 6, and to Laci Babai for the discussions on this topic .  ...  Proof: By the result of Ramsey [7] and Erdős and Szekeres [4] , every n-vertex graph has either a clique on k, or an anti-clique on vertices, if n ≥ k + − 2 k − 1 .  ... 
doi:10.37236/1493 fatcat:twcc5rg2ebcepe3w6rkdnusdmq

Anti-Ramsey Numbers for Graphs with Independent Cycles

Zemin Jin, Xueliang Li
2009 Electronic Journal of Combinatorics  
The anti-Ramsey number $AR(n,{\cal G})$ for ${\cal G}$, introduced by Erdős et al., is the maximum number of colors in an edge coloring of $K_n$ that has no rainbow copy of any graph in ${\cal G}$.  ...  In this paper, we determine the anti-Ramsey number $AR(n,\Omega_2)$, where $\Omega_2$ denotes the family of graphs that contain two independent cycles.  ...  The anti-Ramsey number AR(n, G) for G is the maximum number of colors in an edge coloring of K n that has no rainbow copy of any graph in G.  ... 
doi:10.37236/174 fatcat:lznvgkslsraa7hykaf47gpulhu

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.  ...  Since their introduction by Erdős, Simonovits, and Sós [6] , anti-Ramsey numbers have been well studied from many perspectives. See [10] for an updated list of results in the area.  ...  Anti-Ramsey Numbers Given graphs G and H, the anti-Ramsey number ar(G, H) is defined to be the maximum number of colors k such that there exists a coloring of the edges of G using k colors in which every  ... 
doi:10.20429/tag.2019.060201 fatcat:o7j3a45gbjcstelwt7352z7z2e

Turán's Theorem in the Hypercube

Noga Alon, Anja Krech, Tibor Szabó
2007 SIAM Journal on Discrete Mathematics  
A relationship to anti-Ramsey colorings is also discussed. We discover much less about the Turán-type question which motivated our investigations. Numerous problems and conjectures are raised.  ...  We are motivated by the analogue of Turán's theorem in the hypercube Q n : how many edges can a Q d -free subgraph of Q n have?  ...  We would like to thank an anonymous referee for pointing out reference [2] to us.  ... 
doi:10.1137/060649422 fatcat:lfyj6vsafzhtji6jfwe4haboke

Planar anti-Ramsey numbers for paths and cycles [article]

Yongxin Lan, Yongtang Shi, Zi-Xia Song
2017 arXiv   pre-print
The planar anti-Ramsey number of H, denoted ar__P(n, H), is the maximum number of colors in an edge-coloring of a plane triangulation T∈T_n(H) such that T contains no rainbow copy of H.  ...  Motivated by anti-Ramsey numbers introduced by Erdős, Simonovits and Sós in 1975, we study the anti-Ramsey problem when host graphs are plane triangulations.  ...  anti-Ramsey numbers when host graphs are wheels.  ... 
arXiv:1709.00970v2 fatcat:cd2cbfy32zdxvpxq5vdozagd2y

On degree anti-Ramsey numbers

Shoni Gilboa, Dan Hefetz
2017 European journal of combinatorics (Print)  
In this paper we prove a general upper bound on degree anti-Ramsey numbers, determine the precise value of the degree anti-Ramsey number of any forest, and prove an upper bound on the degree anti-Ramsey  ...  The degree anti-Ramsey number AR_d(H) of a graph H is the smallest integer k for which there exists a graph G with maximum degree at most k such that any proper edge colouring of G yields a rainbow copy  ...  Note that the upper bounds on AR d (C k ) we proved in Theorem 1.4 entail upper bounds on the size anti-Ramsey numbers of cycles.  ... 
doi:10.1016/j.ejc.2016.09.002 fatcat:6e4cnp5cr5gfzlbvukseoh356a

Online and size anti-Ramsey numbers [article]

Maria Axenovich, Kolja Knauer, Judith Stumpp, Torsten Ueckerdt
2013 arXiv   pre-print
The smallest number of edges in a graph any of whose proper edge colorings contains a totally multicolored copy of a graph H is the size anti-Ramsey number AR_s(H) of H.  ...  anti-Ramsey number AR s and local anti-Ramsey number AR loc .  ...  Size Anti-Ramsey Numbers The inequalities in Theorem 1 together with known upper bounds on the local anti-Ramsey number AR loc immediately give upper bounds on AR s , AR o and AR F F .  ... 
arXiv:1311.0539v1 fatcat:k3djj2sidfeplamvkzsealxlya

Online and size anti-Ramsey numbers

Maria Axenovich, Kolja Knauer, Judith Stumpp, Torsten Ueckerdt
2014 Journal of Combinatorics  
The smallest number of edges in a graph any of whose proper edge colorings contains a totally multicolored copy of a graph H is the size anti-Ramsey number ARs(H) of H.  ...  anti-Ramsey number AR s and local anti-Ramsey number AR loc .  ...  Size Anti-Ramsey Numbers The inequalities in Theorem 1 together with known upper bounds on the local anti-Ramsey number AR loc immediately give upper bounds on AR s , AR o and AR F F .  ... 
doi:10.4310/joc.2014.v5.n1.a4 fatcat:e6lbgmq4cjbivmvcs5yn3yqcka

On the anti-Ramsey numbers of linear forests [article]

Tian-Ying Xie, Long-Tu Yuan
2020 arXiv   pre-print
For a fixed graph F, the anti-Ramsey number, AR(n,F), is the maximum number of colors in an edge-coloring of K_n which does not contain a rainbow copy of F.  ...  In this paper, we determine the exact value of anti-Ramsey numbers of linear forests for sufficiently large n, and show the extremal edge-colored graphs.  ...  Very recently, Fang, Győri, Lu and Xiao [4] have given an approximate value of anti-Ramsey number of linear forests and determined the anti-Ramsey number of linear forests whose all components are odd  ... 
arXiv:2003.07541v1 fatcat:ygkfio756jerxfqcbvaaenh7zm

Anti-Ramsey numbers for disjoint copies of graphs

Izolda Gorgol, Agnieszka Görlich
2017 Opuscula Mathematica  
For a graph G and a positive integer n, the anti-Ramsey number ar(n, G) is the maximum number of colors in an edge-coloring of Kn with no rainbow copy of H.  ...  Anti-Ramsey numbers were introduced by Erdős, Simonovits and Sós and studied in numerous papers. Let G be a graph with anti-Ramsey number ar(n, G).  ...  For a graph G and a positive integer n, the anti-Ramsey number ar(n, G) is the maximum number of colors in an edge-coloring of K n with no rainbow copy of G.  ... 
doi:10.7494/opmath.2017.37.4.567 fatcat:hsowur2bjjamrczouub7gbvvqy
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