Reducing Hajَsʹ 4-coloring conjecture to 4-connected graphs

Hajós conjectured that, for any positive integer k, every graph containing no K k + 1 -subdivision is k-colorable. This is true when k ⩽ 3 , and false when k ⩾ 6 . Hajósʹ 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 Hajósʹ conjecture for k = 4 to the conjecture of Seymour that any 5-connected non-planar graph contains a K 5 -subdivision.