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 $\sqrt{\log\log k}$.
From: Licheng Zhang [view email]
[v1]
Thu, 4 Jun 2026 15:36:57 UTC (14 KB)
[v2]
Thu, 23 Jul 2026 13:56:33 UTC (14 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。