Many triangles with few edges

Extremal problems concerning the number of independent sets or complete subgraphs in a graph have been well studied in recent years. Cutler and Radcliffe proved that among graphs with n vertices and maximum degree at most r, where n = a(r + 1) + b and 0 ≤ b ≤ r, aKr+1 ∪ Kb has the maximum number of complete subgraphs, answering a question of Galvin. Gan, Loh and Sudakov conjectured that aKr+1 ∪Kb also maximizes the number of complete subgraphs Kt for each fixed size t ≥3, and proved this for a = 1. Cutler and Radcliffe proved this conjecture for r ≤ 6. We investigate a variant of this problem where we fix the number of edges instead of the number of vertices. We prove that aKr+1 ∪C(b), where C(b) is the colex graph on b edges, maximizes the number of triangles among graphs with m edges and any fixed maximum degree r ≤ 8, where m = a(r+1 2 ) + b and 0 ≤ b < (r+1 2 ). Mathematics Subject Classifications: 05.