(I) We revisit the algorithmic problem of finding all triangles in a graph $G=(V,E)$ with $n$ vertices and $m$ edges. According to a result of Chiba and Nishizeki (1985), this task can be achieved by a combinatorial algorithm running in $O(m \alpha) = O(m^{3/2})$ time, where $\alpha= \alpha(G)$ is the graph arboricity. We provide a new very simple combinatorial algorithm for finding all triangles in a graph and show that is amenable to the same running time analysis. We derive these worst-case bounds from first principles and with very simple proofs that do not rely on classic results due to Nash-Williams from the 1960s. (II) We extend our arguments to the problem of finding all small complete subgraphs of a given fixed size. We show that the dependency on $m$ and $\alpha$ in the running time $O(\alpha^{\ell-2} \cdot m)$ of the algorithm of Chiba and Nishizeki for listing all copies of $K_\ell$, where $\ell \geq 3$, is asymptotically tight. (III) We give improved arboricity-sensitive running times for counting and/or detection of copies of $K_\ell$, for small $\ell \geq 4$. A key ingredient in our algorithms is, once again, the algorithm of Chiba and Nishizeki. Our new algorithms are faster than all previous algorithms in certain high-range arboricity intervals for every $\ell \geq 7$.

翻译：(I) 我们重新审视在顶点数为$n$、边数为$m$的图$G=(V,E)$中寻找所有三角形的算法问题。根据Chiba和Nishizeki (1985)的结果，该任务可通过运行时间为$O(m \alpha) = O(m^{3/2})$的组合算法实现，其中$\alpha= \alpha(G)$为图的树度（arboricity）。我们提出了一种新的极简组合算法用于寻找图中所有三角形，并证明其适用于相同的运行时间分析。我们直接从第一性原理推导出这些最坏情况界，并通过无需依赖Nash-Williams在1960年代经典结论的极简证明完成推导。(II) 我们将论证拓展至寻找固定大小的所有小型完全子图问题。研究表明，对于Chiba-Nishizeki算法中列举所有$K_\ell$（$\ell \geq 3$）副本的$O(\alpha^{\ell-2} \cdot m)$运行时间，其对$m$和$\alpha$的依赖性是渐近紧的。(III) 针对$\ell \geq 4$的小规模情形，我们给出了改进的树度敏感性运行时间，用于$K_\ell$副本的计数与检测。我们算法的关键组件再次基于Chiba-Nishizeki算法。对于每个$\ell \geq 7$，在特定高树度区间内，我们的新算法均快于所有既往算法。