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Ramsey Size Linear Graphs

Published online by Cambridge University Press:  12 September 2008

Paul Erdős ,
R. J. Faudree ,
C. C. Rousseau  and
R. H. Schelp
Paul Erdős
Affiliation:
Mathematical Institute, Hungarian Academy of Sciences, Budapest V, Hungary
R. J. Faudree
Affiliation:
Department of Mathematical Science, Memphis State University, Tenn. 38152USA
C. C. Rousseau
Affiliation:
Department of Mathematical Science, Memphis State University, Tenn. 38152USA
R. H. Schelp
Affiliation:
Department of Mathematical Science, Memphis State University, Tenn. 38152USA
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Abstract

A graph G is Ramsey size linear if there is a constant C such that for any graph H with n edges and no isolated vertices, the Ramsey number r(G, H) ≤ Cn. It will be shown that any graph G with p vertices and q ≥ 2p − 2 edges is not Ramsey size linear, and this bound is sharp. Also, if G is connected and q ≤ p + 1, then G is Ramsey size linear, and this bound is sharp also. Special classes of graphs will be shown to be Ramsey size linear, and bounds on the Ramsey numbers will be determined.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 2 , Issue 4 , December 1993 , pp. 389 - 399
Copyright
Copyright © Cambridge University Press 1993

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References

[1]Bollobás, B. (1978) Extremal Graph Theory, Academic Press, London.Google Scholar
[2]Chvátal, V. (1977) Tree-Complete Graph Ramsey Numbers. J. Graph Theory 1 93.CrossRefGoogle Scholar
[3]Erdős, P. (1965) On some Extremal Problems in Graph Theory. Israel J. Math. 3 113–116.CrossRefGoogle Scholar
[4]Erdős, P. (1965) On an Extremal Problem in Graph Theory. Colloquium Math. 13 251–254.CrossRefGoogle Scholar
[5]Erdős, P., Faudree, R. J., Rousseau, C. C. and Schelp, R. H. (1978) On Cycle-Complete Graph Ramsey Numbers. J. Graph Theory 2 53–64.CrossRefGoogle Scholar
[6]Erdős, P., Faudree, R. J., Rousseau, C. C. and Schelp, R. H. (1987) A Ramsey Problem of Harary on Graphs with Prescribed Size. Discrete Math 67 227–233.CrossRefGoogle Scholar
[7]Erdős, P. and Gallai, T. (1959) On Maximal Paths and Circuits of Graphs. Acta Math. Acad. Sci. Hungar. 10 337–356.CrossRefGoogle Scholar
[8]Erdős, P. and Lovász, L. (1973) Problems and Results on 3-Chromatic Hypergraphs and Some Related Questions. Infinite and Finite Sets 10, Colloquia Mathematica Societatis János Bolyai, Keszthely, Hungary609–628.Google Scholar
[9]Faudree, R. J. (1983) On a Class of Degenerate Extremal Graph Problems. Combinatorica 3 83–93.CrossRefGoogle Scholar
[10]Parsons, T. D. (1975) Ramsey Graphs and Block Designs. Trans. Amer. Math. Soc. 209 33–44.CrossRefGoogle Scholar
[11]Lorimer, P. (1984) The Ramsey Numbers for Stripes and One Complete Graph. J. Graph Theory 8 177–184.CrossRefGoogle Scholar
[12]Sidorenko, A. F. (manuscript) The Ramsey Number of an N-Edge Graph Versus Triangle is at Most 2N + 1.Google Scholar
[13]Simonovits, M. (1983) Extremal Graph Theory. In: Beineke, L. W. and Wilson, R. J. (eds.) Selected Topics in Graph Theory II, Academic Press, New York161–200.Google Scholar
[14]Spencer, J. (1952) Asymptotic Lower Bounds for Ramsey Functions. Discrete Math. 20 69–76.CrossRefGoogle Scholar