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.
Department, Program, or Center
Center for Advancing the Study of CyberInfrastructure
Ars Combinatoria 31 (1991) 65 - 92
RIT – Main Campus