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.
ASJC Scopus subject areas
- Applied Mathematics
- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics