Summary: Testing Triangle-Freeness in General Graphs
In this paper we consider the problem of testing whether
a graph is triangle-free, and more generally, whether it is
H-free, for a fixed subgraph H. The algorithm should
accept graphs that are triangle-free and reject graphs
that are far from being triangle-free in the sense that
a constant fraction of the edges should be removed in
order to obtain a triangle-free graph. The algorithm is
allowed a small probability of error.
This problem has been studied quite extensively in
the past, but the focus was on dense graphs, that is,
when d = (n), where d is the average degree in the
graph and n is the number of vertices. Here we study
the complexity of the problem in general graphs, that
is, for varying d.