# Multicolor Ramsey numbers for paths and cycles

Discussiones Mathematicae Graph Theory (2005)

- Volume: 25, Issue: 1-2, page 57-65
- ISSN: 2083-5892

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topTomasz Dzido. "Multicolor Ramsey numbers for paths and cycles." Discussiones Mathematicae Graph Theory 25.1-2 (2005): 57-65. <http://eudml.org/doc/270242>.

@article{TomaszDzido2005,

abstract = {For given graphs G₁,G₂,...,Gₖ, k ≥ 2, the multicolor Ramsey number R(G₁,G₂,...,Gₖ) is the smallest integer n such that if we arbitrarily color the edges of the complete graph on n vertices with k colors, then it is always a monochromatic copy of some $G_i$, for 1 ≤ i ≤ k. We give a lower bound for k-color Ramsey number R(Cₘ,Cₘ,...,Cₘ), where m ≥ 8 is even and Cₘ is the cycle on m vertices. In addition, we provide exact values for Ramsey numbers R(P₃,Cₘ,Cₚ), where P₃ is the path on 3 vertices, and several values for R(Pₗ,Pₘ,Cₚ), where l,m,p ≥ 2. In this paper we present new results in this field as well as some interesting conjectures.},

author = {Tomasz Dzido},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {edge coloring; Ramsey number},

language = {eng},

number = {1-2},

pages = {57-65},

title = {Multicolor Ramsey numbers for paths and cycles},

url = {http://eudml.org/doc/270242},

volume = {25},

year = {2005},

}

TY - JOUR

AU - Tomasz Dzido

TI - Multicolor Ramsey numbers for paths and cycles

JO - Discussiones Mathematicae Graph Theory

PY - 2005

VL - 25

IS - 1-2

SP - 57

EP - 65

AB - For given graphs G₁,G₂,...,Gₖ, k ≥ 2, the multicolor Ramsey number R(G₁,G₂,...,Gₖ) is the smallest integer n such that if we arbitrarily color the edges of the complete graph on n vertices with k colors, then it is always a monochromatic copy of some $G_i$, for 1 ≤ i ≤ k. We give a lower bound for k-color Ramsey number R(Cₘ,Cₘ,...,Cₘ), where m ≥ 8 is even and Cₘ is the cycle on m vertices. In addition, we provide exact values for Ramsey numbers R(P₃,Cₘ,Cₚ), where P₃ is the path on 3 vertices, and several values for R(Pₗ,Pₘ,Cₚ), where l,m,p ≥ 2. In this paper we present new results in this field as well as some interesting conjectures.

LA - eng

KW - edge coloring; Ramsey number

UR - http://eudml.org/doc/270242

ER -

## References

top- [1] J. Arste, K. Klamroth and I. Mengersen, Three color Ramsey numbers for small graphs, Utilitas Mathematica 49 (1996) 85-96. Zbl0854.05077
- [2] J.A. Bondy and P. Erdös, Ramsey numbers for cycles in graphs, J. Combin. Theory (B) 14 (1973) 46-54, doi: 10.1016/S0095-8956(73)80005-X. Zbl0248.05127
- [3] A. Burr and P. Erdös, Generalizations of a Ramsey-theoretic result of Chvatal, J. Graph Theory 7 (1983) 39-51, doi: 10.1002/jgt.3190070106. Zbl0513.05040
- [4] C. Clapham, The Ramsey number R(C₄,C₄,C₄), Periodica Mathematica Hungarica 18 (1987) 317-318, doi: 10.1007/BF01848105.
- [5] T. Dzido, Computer experience from calculating some 3-color Ramsey numbers (Technical Report of Gdańsk University of Technology ETI Faculty, 2003).
- [6] R. Faudree, A. Schelten and I. Schiermeyer, The Ramsey number R(C₇,C₇,C₇), Discuss. Math. Graph Theory 23 (2003) 141-158, doi: 10.7151/dmgt.1191.
- [7] R.E. Greenwood and A.M. Gleason, Combinatorial relations and chromatic graphs, Canadian J. Math. 7 (1955) 1-7, doi: 10.4153/CJM-1955-001-4. Zbl0064.17901
- [8] T. Łuczak, R(Cₙ,Cₙ,Cₙ) ≤ (4+o(1))n, J. Combin. Theory (B) 75 (1999) 174-187.
- [9] S.P. Radziszowski, Small Ramsey numbers, Electronic J. Combin. Dynamic Survey 1, revision #9, July 2002, http://www.combinatorics.org/. Zbl0953.05048
- [10] P. Rowlison and Y. Yang, On the third Ramsey numbers of graphs with five edges, J. Combin. Math. and Combin. Comp. 11 (1992) 213-222. Zbl0756.05078
- [11] P. Rowlison and Y. Yang, On Graphs without 6-cycles and related Ramsey numbers, Utilitas Mathematica 44 (1993) 192-196. Zbl0789.05070
- [12] D.R. Woodall, Sufficient conditions for circuits in graphs, Proc. London Math. Soc. 24 (1972) 739-755, doi: 10.1112/plms/s3-24.4.739. Zbl0233.05117

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