In a recent paper [Bjarne Toft, A survey of Hadwiger's Conjecture, Institut for Matematik og Datalogi, Odense Universitet, Preprints 1996, Nr. 1] the history of Hadwiger's Conjecture has been explored. It is explained that already in the original paper by Hadwiger [1943] the case k=4 was proved in the following extended form:
Theorem (Hadwiger [1943]). Let G be a graph without K4 as a minor. Then any 3-coloring of an induced cycle C can be extended to a 3-coloring of G.
Hadwiger's proof was short and elegant. Hadwiger [1943] also proved that any graph of minimum degree at least 3 has K4 as a minor. This was the version of Hadwiger's Conjecture for k=4 that Dirac [1952a] proved. Hadwiger's proof was by induction and not particularly elegant, whereas Dirac's proof was an elegant application of longest cycles.
A third extension and proof was presented in the paper [K. Wagner, Bemerkungen zu Hadwiger's Vermutung, Math. Ann. 141 (1960), 570-590].
The above paper by Toft also mentions two more recent very simple proofs of the case k=4 by Woodall [D.R. Woodall, A short proof of a theorem of Dirac's about Hadwiger's Conjecture, J. Graph Theory 16 (1992), 79-80] and by Michael Stiebitz, The Technical University of Ilmenau, Germany.
January 11, 1996 Tommy R. Jensen and Bjarne Toft.
July, 1997 Bjarne Toft