We study the problem of testing $C_k$-freeness ($k$-cycle-freeness) for fixed constant $k > 3$ in graphs with bounded arboricity (but unbounded degrees). In particular, we are interested in one-sided error algorithms, so that they must detect a copy of $C_k$ with high constant probability when the graph is $\epsilon$-far from $C_k$-free. We next state our results for constant arboricity and constant $\epsilon$ with a focus on the dependence on the number of graph vertices, $n$. The query complexity of all our algorithms grows polynomially with $1/\epsilon$. (1) As opposed to the case of $k=3$, where the complexity of testing $C_3$-freeness grows with the arboricity of the graph but not with the size of the graph (Levi, ICALP 2021) this is no longer the case already for $k=4$. We show that $\Omega(n^{1/4})$ queries are necessary for testing $C_4$-freeness, and that $\widetilde{O}(n^{1/4})$ are sufficient. The same bounds hold for $C_5$. (2) For every fixed $k \geq 6$, any one-sided error algorithm for testing $C_k$-freeness must perform $\Omega(n^{1/3})$ queries. (3) For $k=6$ we give a testing algorithm whose query complexity is $\widetilde{O}(n^{1/2})$. (4) For any fixed $k$, the query complexity of testing $C_k$-freeness is upper bounded by ${O}(n^{1-1/\lfloor k/2\rfloor})$. Our $\Omega(n^{1/4})$ lower bound for testing $C_4$-freeness in constant arboricity graphs provides a negative answer to an open problem posed by (Goldreich, 2021).

翻译：我们研究在具有有界树图（但无界度数）的图中，针对固定常数$k > 3$的$C_k$-无环性（$k$-环无环性）测试问题。特别关注单侧错误算法，要求当图与$C_k$-无环性相距$\epsilon$时，算法能以高常数概率检测出$C_k$副本。下面针对常数树图和常数$\epsilon$给出结果，重点分析对图顶点数$n$的依赖关系。所有算法的查询复杂度均关于$1/\epsilon$呈多项式增长。(1) 与$k=3$的情况不同——此时测试$C_3$-无环性的复杂度随图树图增长而非图规模增长（Levi, ICALP 2021），当$k=4$时该结论已不成立。我们证明测试$C_4$-无环性需要$\Omega(n^{1/4})$次查询，且$\widetilde{O}(n^{1/4})$次查询即足够。$C_5$的界相同。(2) 对每个固定$k \geq 6$，测试$C_k$-无环性的任意单侧错误算法必须执行$\Omega(n^{1/3})$次查询。(3) 对于$k=6$，我们给出查询复杂度为$\widetilde{O}(n^{1/2})$的测试算法。(4) 对任意固定$k$，测试$C_k$-无环性的查询复杂度上界为${O}(n^{1-1/\lfloor k/2\rfloor})$。我们关于常数树图中测试$C_4$-无环性的$\Omega(n^{1/4})$下界，对（Goldreich, 2021）提出的开放问题给出了否定答案。