Complete graph immersions in dense graphs

Then, we study the class of all graphs with independence number less than three, which are graphs of interest for Hadwiger's Conjecture. We study such graphs for the immersion-analog. If Abu-Khzam and Langston's conjecture is true for this class of graphs, then an easy argument shows that every graph of independence number less than 3 contains Kn2 as an immersion. We show that the converse is also true. That is, if every graph with independence number less than 3 contains an immersion of Kn2, then Abu-Khzam and Langston's conjecture is true for this class of graphs. Furthermore, we show that every graph of independence number less than 3 has an immersion of Kn3.