Immersing small complete graphs

Abstract

Following in the spirit of the Hadwiger and Hajo ́s conjectures, Abu-Khzam and Lang- ston have conjectured that every k-chromatic graph contains an immersion of Kk. They proved this for k ≤ 4. Much before that, Lescure and Meyniel [10] obtained a stronger result that included also the values k = 5 and 6, by proving that every simple graph of minimum degree k − 1 contains an immersion of Kk. They noted that they also have a proof of the same result for k = 7 but have not published it due to the length of the proof. We give a simple proof of this result. This, in particular, proves the conjecture of Abu-Khzam and Langston for every k ≤ 7.