Complete graph immersions in dense graphs

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 $K_{\left\lceil\frac{n}{3}\right\rceil}$.

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