Testing C_k-Freeness in Bounded-Arboricity Graphs

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 ϵ-far from C_k-free. We next state our results for constant arboricity and constant ϵ with a focus on the dependence on the number of graph vertices, n. The query complexity of all our algorithms grows polynomially with 1/ϵ. 1. As opposed to the case of k = 3, where the complexity of testing C3-freeness grows with the arboricity of the graph but not with the size of the graph (Levi, ICALP 2021)¹ this is no longer the case already for k = 4. We show that Ω(n¹/⁴) queries are necessary for testing C4-freeness, and that O^e(n¹/⁴) are sufficient. The same bounds hold for C5. 2. For every fixed k ≥ 6, any one-sided error algorithm for testing C_k-freeness must perform Ω(n¹/³) queries. 3. For k = 6 we give a testing algorithm whose query complexity is O^e(n¹/²). 4. For any fixed k, the query complexity of testing C_k-freeness is upper bounded by O(n¹⁻¹/⌊k/2^⌋). The last upper bound builds on another result in which we show that for any fixed subgraph F, the query complexity of testing F-freeness is upper bounded by O(n¹⁻¹/ℓ(F)), where ℓ(F) is a parameter of F that is always upper bounded by the number of vertices in F (and in particular is k/2 in C_k for even k). We extend some of our results to bounded (non-constant) arboricity, where in particular, we obtain sublinear upper bounds for all k. Our Ω(n¹/⁴) lower bound for testing C4-freeness in constant arboricity graphs provides a negative answer to an open problem posed by (Goldreich, 2021).