Abstract
Let β ( G ) denote the minimum number of edges to be removed from a graph G to make it bipartite. For each 3-chromatic graph F we determine a parameter ξ ( F ) such that for each F-free graph G on n vertices with minimum degree δ ( G ) ⩾ 2 n / ( ξ ( F ) + 2 ) + o ( n ) we have β ( G ) = o ( n 2 ) , while there are F-free graphs H with δ ( H ) ≥ ⌊ 2 n / ( ξ ( F ) + 2 ) ⌋ for which β ( H ) = Ω ( n 2 ) .
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