Minimum degree condition forcing complete graph immersion

An immersion of a graph $H$ into a graph $G$ is a one-to-one mapping $f:V(H) \to V(G)$andacollectionofedge-disjointpathsin$G$,oneforeachedgeof$H$,suchthatthepath$P_{uv}$correspondingtoedge$uv$hasendpoints$f(u)$and$f(v)$\mathrm{.}Theimmersionisstrongifthepaths$P_{uv}$ are internally disjoint from $f(V(H))$. It is proved that for every positive integer $t$, every simple graph of minimum degree at least $200t$ contains a strong immersion of the complete graph $K_t$. For dense graphs one can say even more. If the graph has order $n$ and has $2c{n}^2$ edges, then there is a strong immersion of the complete graph on at least $c^{2}n$ vertices in $G$ in which each path $P_{uv}$ is of length 2. As an application of these results, we resolve a problem raised by Paul Seymour by proving that the line graph of every simple graph with average degree $d$ has a clique minor of order at least $c{d}^{3\mathrm{/}2}$, where $c>0$ is an absolute constant. For small values of $t$, $1\le t\le 7$, every simple graph of minimum degree at least $t-1$ contains an immersion of $K_t$ (Lescure and Meyniel, DeVos et al.). We provide a general class of examples showing that this does not hold when $t$ is large.