  In this article we consider the relationship between vertex coloring and the
immersion order. Specifically, a conjecture proposed by Abu-Khzam and Langston
in 2003, which says that the complete graph with $t$ vertices can be immersed
in any $t$-chromatic graph, is studied.
  First, we present a general result about immersions and prove that the
conjecture holds for graphs whose complement does not contain any induced cycle
of length four and also for graphs having the property that every set of five
vertices induces a subgraph with at least six edges.
  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
$K_{\left\lceil\frac{n}{2}\right\rceil}$ 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 $K_{\left\lceil\frac{n}{2}\right\rceil}$,
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 $K_{\left\lceil\frac{n}{3}\right\rceil}$.
