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Degree sum for a triangle in a graph

Let (n; e) denote the class of graphs on n vertices and e edges. Define f(n, e) = min max{Σd(vi):{v1, v2,v3} induces a triangle in G}, where the maximum is taken over all triangles in the graph G and the minimum is taken over all G in (n; e). From Turan's theorem, f(n, e) = 0 if e ⩽ n2/4; otherwise f(n,e) > 0. Bollobas and Erdos [1] asked to determine the function f(n,e) for e > n2/4. Edwards [3] proved that f(n,e) ⩾ 6e/n for e ≥ n2/3. In this paper, we consider the remaining case, namely, n2/4 21n/16, which improves a result of Erdos and Laskar [4] that f(n, ⌊n2/4⌋ + 1) > (1 + e)n for some positive constant e. Furthermore, if e ⩾ 0.26n2, we obtain a better result.