Abstract

We prove that e(3, k + 1, n) >= 6n - 13k where e(3, k + 1, n) is the minimum number of edges in any triangle-free graph on n vertices with no independent set of size k + 1. to achieve this we first characterize all such graphs with exactly e(3, k + 1, n) edges for n <= 3k. These results yield some sharp lower bounds for the independence ratio for trianagle-free graphs. In particular, the exact value of the minimal independence ratio for graphs with average degree 4 is shown in be 4/13. A slight improvement to the general upper bound for the classical Ramsey R(3,k) numbers is also obtained.

Publication Date

1991

Comments

ISSN:0381-7032 Note: imported from RIT’s Digital Media Library running on DSpace to RIT Scholar Works in February 2014.

Document Type

Article

Department, Program, or Center

Center for Advancing the Study of CyberInfrastructure

Campus

RIT – Main Campus

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