Skip to main content

Ramsey Size Linear Graphs

  • Paul Erdős (a1), R. J. Faudree (a2), C. C. Rousseau (a2) and R. H. Schelp (a2)

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 qp + 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.

Hide All
[1]Bollobás, B. (1978) Extremal Graph Theory, Academic Press, London.
[2]Chvátal, V. (1977) Tree-Complete Graph Ramsey Numbers. J. Graph Theory 1 93.
[3]Erdős, P. (1965) On some Extremal Problems in Graph Theory. Israel J. Math. 3 113116.
[4]Erdős, P. (1965) On an Extremal Problem in Graph Theory. Colloquium Math. 13 251254.
[5]Erdős, P., Faudree, R. J., Rousseau, C. C. and Schelp, R. H. (1978) On Cycle-Complete Graph Ramsey Numbers. J. Graph Theory 2 5364.
[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 227233.
[7]Erdős, P. and Gallai, T. (1959) On Maximal Paths and Circuits of Graphs. Acta Math. Acad. Sci. Hungar. 10 337356.
[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, Hungary609628.
[9]Faudree, R. J. (1983) On a Class of Degenerate Extremal Graph Problems. Combinatorica 3 8393.
[10]Parsons, T. D. (1975) Ramsey Graphs and Block Designs. Trans. Amer. Math. Soc. 209 3344.
[11]Lorimer, P. (1984) The Ramsey Numbers for Stripes and One Complete Graph. J. Graph Theory 8 177184.
[12]Sidorenko, A. F. (manuscript) The Ramsey Number of an N-Edge Graph Versus Triangle is at Most 2N + 1.
[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 York161200.
[14]Spencer, J. (1952) Asymptotic Lower Bounds for Ramsey Functions. Discrete Math. 20 6976.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed