Complete graph immersions and minimum degree

TL;DR

This paper proves that any simple graph with a sufficiently high minimum degree contains a strong immersion of a complete graph, improving previous bounds significantly.

Contribution

It establishes a new lower bound on the minimum degree needed for strong immersions of complete graphs in simple graphs.

Findings

Graphs with minimum degree ≥ 11t+7 contain a strong K_t immersion.

Improves previous minimum degree bound of 200t for such immersions.

Advances understanding of graph immersions and their degree conditions.

Abstract

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 P_{uv} corresponding to the edge uv has endpoints phi(u) and phi(v). The immersion is strong if the paths P_{uv} 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 K_t. This improves on previously known bound of minimum degree at least 200t obtained by DeVos et al.