Many Triangles with Few Edges

Abstract

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 $0 \le b \le r$, $aK_{r+1}\cup K_b$ has the maximum number of complete subgraphs, answering a question of Galvin. Gan, Loh, and Sudakov conjectured that $aK_{r+1}\cup K_b$ also maximizes the number of complete subgraphs $K_t$ for each fixed size $t \ge 3$, and proved this for $a = 1$. Cutler and Radcliffe proved this conjecture for $r \le 6$. We investigate a variant of this problem where we fix the number of edges instead of the number of vertices. We prove that $aK_{r+1}\cup \mathcal{C}(b)$, where $\mathcal{C}(b)$ is the colex graph on $b$ edges, maximizes the number of triangles among graphs with $m$ edges and any fixed maximum degree $r\le 8$, where $m = a \binom{r+1}{2} + b$ and $0 \le b < \binom{r+1}{2}$.