Immersing complete digraphs

TL;DR

This paper investigates conditions under which complete digraphs can be immersed in simple digraphs, establishing degree thresholds for Eulerian cases and providing counterexamples for non-Eulerian digraphs, along with a related graph construction.

Contribution

It proves degree conditions for immersing complete digraphs in Eulerian digraphs and constructs examples showing limitations in non-Eulerian cases, also simplifying existing graph constructions.

Findings

Eulerian digraphs with minimum degree ≥ t(t-1) contain the complete digraph immersion.

For t ≤ 4, minimum degree ≥ t-1 suffices for immersion.

Existence of non-Eulerian digraphs with high degrees that do not contain the complete digraph on 3 vertices.

Abstract

We consider the problem of immersing the complete digraph on t vertices in a simple digraph. For Eulerian digraphs, we show that such an immersion always exists whenever minimum degree is at least t(t-1), and for t at most 4 minimum degree at least t-1 suffices. On the other hand, we show that there exist non-Eulerian digraphs with all vertices of arbitrarily high in- and outdegree which do not contain an immersion of the complete digraph on 3 vertices. As a side result, we obtain a construction of digraphs with large outdegree in which all cycles have odd length, simplifying a former construction of such graphs by Thomassen.