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) → V (G) and a collection of edge-disjoint paths in G, one for each edge of H , such that the path Puv corresponding to edge uv has endpoints f(u) and f(v). The immersion is strong if the paths Puv 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 Kt. 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 cn vertices in G in which each path Puv 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, 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 Kt (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. Supported in part by an NSERC Discovery Grant (Canada) and a Sloan Fellowship. Supported in part by the grant GA201/09/0197 of Czech Science Foundation and by Institute for Theoretical Computer Science (ITI), project 1M0021620808 of Ministry of Education of Czech Republic. Supported in part by a Simons Fellowship. Supported by an NSERC Postdoctoral fellowship. Supported in part by an NSERC Discovery Grant (Canada), by the Canada Research Chair program, and by the Research Grant P1–0297 of ARRS (Slovenia). On leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia. Postdoctoral fellowship at Simon Fraser University, Burnaby.