A graph is defined to be a collection of points (vertices) and lines (edges) between vertices. A cycle in a graph is a path that returns to where it started without repeating any vertices other than the last one. In 1995 the Hungarian mathematicians Paul Erdos and Andras Gyarfas suggested that every graph with at least three edges connected to each vertex must contain a cycle with length a power of 2. I will discuss progress made on this conjecture, with a focus on a class of graphs known as cubic bipartite graphs. I will also include a result on the number of edges required to guarantee a cycle of length 4 in any graph.