Hajós conjectured that, for any positive integer k, every graph containing no K_(k+1)-subdivision is k-colorable. This is true when k is at most three, and false when k exceeds six. Hajós' conjecture remains open for k=4,5. We will first present some known results on Hajós' conjecture. Then we derive a result on the structure of 2-connected graphs with no cycle through three specified vertices. This result will then be used for the proof of the main result of this thesis. We show that any possible counterexample to Hajós' conjecture for k=4 with minimum number of vertices must be 4-connected. This is a step in an attempt to reduce Hajós' conjecture for k=4 to the conjecture of Seymour that any 5-connected non-planar graph contains a K_5-subdivision.