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{Σ   i=1 3d ( v i ):{ v 1 , v 2 , v 3 } 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 Turán's theorem, f ( n, e ) = 0 if e ⩽ n 2 /4; otherwise f ( n , e ) > 0. Bollobás and Erdös [1] asked to determine the function f ( n , e ) for e > n 2 /4. Edwards [3] proved that f ( n , e ) ⩾ 6 e / n for e ≥ n 2 /3. In this paper, we consider the remaining case, namely, n 2 /4 < e < n 2 /3. A construction described in [4] shows that f ( n , e ) < 4 √3¯ e ‐ 2 n + 5. We prove that f ( n , e ) ≥ 21 e /4 n . In particular, f ( n , ⌊ n 2 /4⌋ + 1) > 21 n /16, which improves a result of Erdös and Laskar [4] that f ( n , ⌊ n 2 /4⌋ + 1) > (1 + ε) n for some positive constant ε. Furthermore, if e ⩾ 0.26 n 2 , we obtain a better result.