  In 2022, Jinha Kim proved a conjecture by Engstr\"om that states the
independence complex of a graph with no induced cycle of length divisible by 3
is either contractible or homotopy equivalent to a sphere. We give criteria for
when the independence complex of a ternary graph is contractible, and describe
the dimension of the sphere when it is not. We then apply our results to
describe the multigraded betti numbers of the edge ideal of a ternary graph. In
particular, we show that the regularity and depth of edge ideals of a ternary
graph $G$ are equal if and only if the independence complex of $G$ is not
contractible. Finally, we apply our results to partially recover and generalize
recent results on the depth of edge ideals of some unicyclic graphs.
