On the minimal degree implying equality of the largest triangle-free and bipartite subgraphs

Erdős posed the problem of finding conditions on a graph G that imply t ( G ) = b ( G ) , where t ( G ) is the largest number of edges in a triangle-free subgraph and b ( G ) is the largest number of edges in a bipartite subgraph. Let δ c be the least number so that any graph G on n vertices with minimum degree δ c n has t ( G ) = b ( G ) . Extending results of Bondy, Shen, Thomassé and Thomassen we show that 0.75 ⩽ δ c < 0.791 .