On the maximum number of edges of k-cacti

Abstract

A cactus is a graph in which every edge lies on at most one cycle. In 2024, Zhang and Huang generalized this concept to the $k$-cactus, defined as a graph in which every edge lies on at most $k$ cycles. It is known that any cactus on $n$ vertices has at most $\lfloor \frac{3}{2}(n-1)\rfloor$ edges. However, the upper bound on the size of $k$-cacti was known only for $k\le 4$. In this note we consider general $k$. 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 $\sqrt{\log \log k}$.