  We study the problem of testing triangle freeness in the general graph model.
This problem was first studied in the general graph model by Alon et al. (SIAM
J. Discret. Math. 2008) who provided both lower bounds and upper bounds that
depend on the number of vertices and the average degree of the graph. Their
bounds are tight only when $d_{\rm max} = O(d)$ and $\bar{d} \leq \sqrt{n}$ or
when $\bar{d} = \Theta(1)$, where $d_{\rm max}$ denotes the maximum degree and
$\bar{d}$ denotes the average degree of the graph. In this paper we provide
bounds that depend on the arboricity of the graph and the average degree. As in
Alon et al., the parameters of our tester is the number of vertices, $n$, the
number of edges, $m$, and the proximity parameter $\epsilon$ (the arboricity of
the graph is not a parameter of the algorithm). The query complexity of our
tester is $\tilde{O}(\Gamma/\bar{d} + \Gamma)\cdot poly(1/\epsilon)$ on
expectation, where $\Gamma$ denotes the arboricity of the input graph (we use
$\tilde{O}(\cdot)$ to suppress $O(\log \log n)$ factors). We show that for
graphs with arboricity $O(\sqrt{n})$ this upper bound is tight in the following
sense. For any $\Gamma \in [s]$ where $s= \Theta(\sqrt{n})$ there exists a
family of graphs with arboricity $\Gamma$ and average degree $\bar{d}$ such
that $\Omega(\Gamma/\bar{d} + \Gamma)$ queries are required for testing
triangle freeness on this family of graphs. Moreover, this lower bound holds
for any such $\Gamma$ and for a large range of feasible average degrees.
