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arXiv · 2606.06298

On the maximum number of edges of k-cacti

Yuanqiu Huang·Licheng Zhang
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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 $\sqrt{\log\log k}$.

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Yuanqiu Huang, Licheng Zhang. 2026-06-04. On the maximum number of edges of k-cacti. https://arxiv.org/abs/2606.06298

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