  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).
