The density of k-cacti via excluding minors

A \emph{$k$-cactus} generalizes forests and cacti by allowing each edge to lie on at most $k$ cycles. The maximum number of edges is classical for forests and cacti, but for $k$-cacti was known only for $k\le 4$. In this note we treat general $k$. The key idea is that bounding the cycles through each edge forces a $k$-cactus to exclude a large complete minor; in particular, the class of $k$-cacti is minor-closed. From this we prove that every $n$-vertex $k$-cactus has $O\!\left(\frac{\log k}{\sqrt{\log\log k}}\,n\right)$ edges for all sufficiently large $k$, and a construction shows this is optimal up to a factor of $\sqrt{\log\log k}$.