Immersion of complete digraphs in Eulerian digraphs

TL;DR

This paper proves that every Eulerian digraph with a certain minimum out-degree can immerse a large complete digraph, advancing understanding of graph immersions and answering a specific open question.

Contribution

It establishes a lower bound on the size of complete digraphs immersed in Eulerian digraphs with given out-degree, resolving an open problem in graph theory.

Findings

Eulerian digraphs with minimum out-degree t immerse a complete digraph of size proportional to t

Provides a constructive proof for the immersion of complete digraphs in Eulerian digraphs

Answers an open question posed by DeVos, Mcdonald, Mohar, and Scheide

Abstract

A digraph $G$ \emph{immerses} a digraph $H$ if there is an injection $f : V(H) \to V(G)$ and a collection of pairwise edge-disjoint directed paths $P_{uv}$, for $uv \in E(H)$, such that $P_{uv}$ starts at $u$ and ends at $v$. We prove that every Eulerian digraph with minimum out-degree $t$ immerses a complete digraph on $\Omega(t)$ vertices, thus answering a question of DeVos, Mcdonald, Mohar, and Scheide.