A minimum degree condition forcing complete graph immersion

Abstract

An immersion of a graph H into a graph G is a one-to-one mapping f: V (H) → V (G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P _uv corresponding to edge uv has endpoints f(u) and f(v). The immersion is strong if the paths P _uv are internally disjoint from f(V (H)). It is proved that for every positive integer Ht, 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 2cn 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 cd ^3/2, where c>0 is an absolute constant.

For small values of t, 1≤t≤7, every simple graph of minimum degree at least t−1 contains an immersion of K _t (Lescure and Meyniel [13], DeVos et al. [6]). We provide a general class of examples showing that this does not hold when t is large.