Complete graph immersions and minimum degree

Abstract

AbstractAn immersion of a graph H in another graph G is a one‐to‐one mapping and a collection of edge‐disjoint paths in G, one for each edge of H, such that the path corresponding to the edge has endpoints and . The immersion is strong if the paths are internally disjoint from . We prove that every simple graph of minimum degree at least contains a strong immersion of the complete graph . This improves on previously known bound of minimum degree at least 200t obtained by DeVos et al. 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 ‐immersion is ‐colorable.