An Anti-Ramsey Theorem on Diamonds

“…⇔ n 2 − 11n + 28 ≤ 0 (7) In [8] projective planes have been used to construct an infinite family of graphs with girth 6 (i.e. {C 3 , C 4 , C 5 }-free) having Ω(n Montellano-Ballesteros [9] has shown an upper bound for the rainbow number of the diamond.…”

Section: Rainbow Numbers For the Bullmentioning

confidence: 99%

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Rainbow numbers for graphs with cyclomatic number at most two

Schiermeyer,

Sotak

2012

Preprint

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For a given graph H and n ≥ 1, let f (n, H) denote the maximum number m for which it is possible to colour the edges of the complete graph K n with m colours in such a way that each subgraph H in K n has at least two edges of the same colour. Equivalently, any edge-colouring of K n with at least rb(n, H) = f (n, H)+1 colours contains a rainbow copy of H. The numbers f (n, H) and rb(K n , H) are called anti-ramsey numbers and rainbow numbers, respectively.In this paper we will classify the rainbow number for a given graph H with respect to its cyclomatic number. Let H be a graph of order p ≥ 4 and cyclomatic number v(H) ≥ 2. Then rb(K n , H) cannot be bounded from above by a function which is linear in n. If H has cyclomatic number v(H) = 1, then rb(K n , H) is linear in n.We will compute all rainbow numbers for the bull B, which is the unique graph with 5 vertices and degree sequence (1, 1, 2, 3, 3). Furthermore, we will compute some rainbow numbers for the diamond D = K 4 − e, for K 2,3 , and for the house H = P 5 .

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“…Rainbow numbers for cycles with one or two additional edges have been studied in [4,5,8,10]. Jendrol' et al in [7] investigated the problem of finding the minimum number of colors that forces the existence of a rainbow face in any edge coloring of a fixed plane graph.…”

Section: Introductionmentioning

confidence: 99%

Rainbow Numbers for Cycles in Plane Triangulations

Horňák

Jendroľ

Schiermeyer

et al. 2014

Journal of Graph Theory

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Abstract:In the article, the existence of rainbow cycles in edge colored plane triangulations is studied. It is shown that the minimum number rb(T n , C 3 ) of colors that force the existence of a rainbow C 3 in any n-vertex plane triangulation is equal to 3n−4 2 . For k ≥ 4 a lower bound and for k ∈ {4, 5} an upper bound of the number rb(T n , C k ) is determined. C 2014 Wiley Periodicals, Inc. J. Graph Theory 78: [248][249][250][251][252][253][254][255][256][257] 2015

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An Anti-Ramsey Theorem on Diamonds

Cited by 6 publications

References 3 publications

Rainbow generalizations of Ramsey theory - a dynamic survey

Rainbow generalizations of Ramsey theory - a dynamic survey

Rainbow numbers for graphs with cyclomatic number at most two

Rainbow Numbers for Cycles in Plane Triangulations

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