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Local Anti-Ramsey Numbers of Graphs

MARIA AXENOVICH, TAO JIANG, ZSOLT TUZA
2003 Combinatorics, probability & computing  
We study properly coloured subgraphs and rainbow subgraphs forced in edge-colourings of complete graphs in which each vertex is incident to a large number of colours.  ...  A subgraph H in an edge-colouring is properly coloured if incident edges of H are assigned different colours, and H is rainbow if no two edges of H are assigned the same colour.  ...  In this paper, we consider the local variation of the anti-Ramsey problem.  ... 
doi:10.1017/s0963548303005868 fatcat:vfiyiaxec5hqnfx6vwj2wqyxdi

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.  ...  This number in offline and online setting is investigated here.  ...  anti-Ramsey number AR s and local anti-Ramsey number AR loc .  ... 
doi:10.4310/joc.2014.v5.n1.a4 fatcat:e6lbgmq4cjbivmvcs5yn3yqcka

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.  ...  This number in offline and online setting is investigated here.  ...  anti-Ramsey number AR s and local anti-Ramsey number AR loc .  ... 
arXiv:1311.0539v1 fatcat:k3djj2sidfeplamvkzsealxlya

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  ...  Within the class of graphs in which each vertex is incident to many colors, the anti-Ramsey problem was studied in [18] under the title of local anti-Ramsey numbers.  ...  The idea of anti-Ramsey numbers for cliques was extended in [27] to coloring in rounds.  ... 
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.  ...  Acknowledgement The authors would like to thank Maria Axenovich, Tao Jiang and the referee for helpful comments and corrections on the initial publication of this work.  ... 
doi:10.20429/tag.2014.000101 fatcat:zawfrrlt4jc4fbtiebsbpprsli

On degree anti-Ramsey numbers

Shoni Gilboa, Dan Hefetz
2017 European journal of combinatorics (Print)  
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  ...  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  ...  Local anti-Ramsey numbers are discussed in [3] as a special case of a broader notion.  ... 
doi:10.1016/j.ejc.2016.09.002 fatcat:6e4cnp5cr5gfzlbvukseoh356a

Page 6741 of Mathematical Reviews Vol. , Issue 2004i [page]

2004 Mathematical Reviews  
Summary: “Certain generalizations of the Ramsey and anti- Ramsey numbers have been introduced and studied in the past few years.  ...  The rainbow Ramsey number, also called the constrained Ramsey number, RR(G,,G2) of two graphs G; and G> is the minimum integer N such that any edge-coloring of the complete graph Ky with any number of  ... 

Size and Degree Anti-Ramsey Numbers

Noga Alon
2015 Graphs and Combinatorics  
The size anti-Ramsey number of H, denoted by AR s (H), is the smallest number of edges in a graph G such that any of its proper edge-colorings contains a rainbow copy of H.  ...  This settles a problem of Axenovich, Knauer, Stumpp and Ueckerdt. The proof is probabilistic and suggests the investigation of a related notion, which we call the degree anti-Ramsey number of a graph.  ...  Part of this work was done during the Japan Conference on Graph Theory and Combinatorics, which took place in Nihon University, Tokyo, in May, 2014.  ... 
doi:10.1007/s00373-015-1583-9 fatcat:szdb4aspvvh6rdlkw3ybbkf7cm

Page 1720 of Mathematical Reviews Vol. 50, Issue 6 [page]

1975 Mathematical Reviews  
Presia, A theorem on the theory of the four-colour problem (pp. 1143-1148); G. B. Purdy, Planarity of two-point universal graphs (pp. 1149-1157); R. Rado,” Anti-Ramsey theorems (pp.  ...  Sds, Anti- Ramsey theorems (pp. 633-643); E. Etourneau, Existence and connectivity of planar graphs having 12 vertices of degree 5 and n-12 vertices of degree 6 (pp. 645-655); R. J. Faudree and R. H.  ... 

Rainbows in the Hypercube

Maria Axenovich, Heiko Harborth, Arnfried Kemnitz, Meinhard Möller, Ingo Schiermeyer
2007 Graphs and Combinatorics  
In this paper we start the investigation of this anti-Ramsey problem by providing bounds on f (Q n , Q k ) which are asymptotically tight for k = 2 and by giving some exact results.  ...  An edge coloring of a graph H is called rainbow if no two edges of H have the same color.  ...  There is a number of results on anti-Ramsey numbers under different local constraints and on anti-Ramsey numbers of the structures other than graphs, such as posets, integers and so on.  ... 
doi:10.1007/s00373-007-0691-6 fatcat:lfxfp43fbjfgnexi54wmo5ediq

Contents

2007 Discrete Mathematics  
Huang New upper bound formulas with parameters for Ramsey numbers 760 K.-I. Kawarabayashi and A.  ...  Wanke A local characterization of bounded clique-width for line graphs 756 Y. Huang, Y. Wang, W. Sheng, J. Yang, K. Zhang and J.  ... 
doi:10.1016/s0012-365x(07)00008-8 fatcat:g2ns66lzezewzfytkhjrr6tasq

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.  ...  The number of colors is tight up to a constant factor, as it turns out that a similar coloring with d+1 2 + 1 colors is not possible. The corresponding question for vertices is also considered.  ...  A coloring c of the edges of graph G is called H-anti-Ramsey if the restriction of c to any subgraph H 0 ⊆ G, H 0 ∼ = H, is not rainbow.  ... 
doi:10.1137/060649422 fatcat:lfyj6vsafzhtji6jfwe4haboke

Page 2850 of Mathematical Reviews Vol. , Issue 86g [page]

1986 Mathematical Reviews  
The authors determine the reduced Ramsey numbers of the ten graphs of order four or less with no isolates.  ...  Mader (Hannover) Babai, Laszlé6 (H-EOTVO-A) 86g:05063 An anti-Ramsey theorem. Graphs Combin. 1 (1985), no. 1, 23-28.  ... 

On Homomorphism Graphs [article]

Sebastian Brandt, Yi-Jun Chang, Jan Grebík, Christoph Grunau, Václav Rozhoň, Zoltán Vidnyánszky
2021 arXiv   pre-print
In particular, we show that for Δ>2 it is impossible to give a simple characterization of acyclic Δ-regular Borel graphs with Borel chromatic number at most Δ: such graphs form a Σ^1_2-complete set.  ...  The motivation for the construction comes from the adaptation of this method to the LOCAL model of distributed computing.  ...  Let us denote by χ wpr−∆ 1 2 (H) the weakly provably ∆ 1 2 -chromatic number of H (see Section 2). Theorem 1.1. Let H be a locally countable Borel graph.  ... 
arXiv:2111.03683v1 fatcat:exhbxopan5e6hauam365k7ww5i

Ramsey games

József Beck
2002 Discrete Mathematics  
This is a partly survey, partly new results paper about Ramsey games. Ramsey games belong to the wider class of positional games.  ...  In Section 1 we brie y recall the basic concepts and results of positional games in general, and apply them to the particular case of Ramsey games.  ...  Ramsey Graph Game The board of the game is the complete graph K s with s vertices.  ... 
doi:10.1016/s0012-365x(01)00224-2 fatcat:ty6tqyllong6hcy65yvbrblwvi
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