Reducing Hajós' 4-coloring conjecture to 4-connected graphs

Hajos conjectured that, for any positive integer k, every graph containing no Kk+1-subdivision is k-colorable. This is true when k ≤ 3, and false when k ≥ 6. Hajos' conjecture remains open for k = 4, 5. In this paper, we show that any possible counterexample to this conjecture for k = 4 with minimum number of vertices must be 4-connected. This is a step in an attempt to reduce Hajos' conjecture for k = 4 to the conjecture of Seymour that any 5-connected non-planar graph contains a K5-subdivision.