Abstrakt
An immersion of a graph H in another graph G is a one-to-one mapping phi: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 the edge uv has endpoints phi(u) and phi(v). The immersion is strong if the paths Puv are internally disjoint from phi(V(H)).
We prove that every simple graph of minimum degree at least 11t+7 contains a strong immersion of the complete graph Kt. This improves on previously known bound of minimum degree at least 200t obtained by DeVos etal.
Our result supports a conjecture of Lescure and Meyniel(also independently proposed by Abu-Khzam and Langston), which is the analogue of famous Hadwiger's conjecture for immersions and says that every graph without a Kt-immersion is (t-1)-colorable.