An immersion of a graph H in another graph G is a one‐to‐one mapping ϕ: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 ϕ(u) and ϕ(v). The immersion is strong if the paths Puv are internally disjoint from ϕ(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 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 Kt‐immersion is (t−1)‐colorable.
Journal of Graph Theory – Wiley
Published: Jan 1, 2018
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