Immersion of transitive tournaments in digraphs with large minimum outdegree

Abstract

We prove the existence of a function h(k) such that every simple digraph with minimum outdegree greater than h(k) contains an immersion of the transitive tournament on k vertices. This solves a conjecture of Devos, McDonald, Mohar and Scheide.In this note, all digraphs are without loops. A digraph D is simple if there is at most one arc from x to y for any x, y ∈ V (D). Note that arcs in opposite directions are allowed. The multiplicity of a digraph D is the maximum number of parallel arcs in the same direction in D. We say that a digraph D contains an immersion of a digraph H if the vertices of H are mapped to distinct vertices of D, and the arcs of H are mapped to directed paths joining the corresponding pairs of vertices of D, in such a way that these paths are pairwise arc-disjoint. If the directed paths are vertex-disjoint, we say that D contains a subdivision of H.Understanding the necessary conditions for graphs to contain a subdivision of a clique is a very natural and well-studied question. One of the most important examples is the following result by Mader [6]:Theorem 1 ([6]). For every k ≥ 1, there exists an integer f (k) such that every graph with minimum degree greater than f (k) contains a subdivision of K k .

Isomorphism testing of $k$-spanning tournaments is Fixed Parameter Tractable

“…We claim that if X is a tournament with k(X) ≤ k and colored by the Weisfeiler-Leman algorithm (see Subsection 2.4), then X is O(k 3 )-spanning. Indeed, for every vertex-color class ∆ of X, the induced subdigraph X ∆ is a regular tournament with k(X ∆ ) ≤ k(X) ≤ k. By [7], this yields |∆| = O(k 3 ). Thus, the maximum valency of each arc-color class of X is at most O(k 3 ).…”

“…Tournaments with bounded immersion. Recall that a tournament Y is immersed in a tournament X if the vertices of Y are mapped to distinct vertices of X and the arcs of Y are mapped to directed paths joining the corresponding pairs of vertices of X, in such a way that these paths are pairwise arc-disjoint (see, e.g., [7]). Given a tournament X, denote by k(X) the maximal integer k such that a transitive tournament 1 with k vertices is immersed in X.…”

Immersion of complete digraphs in Eulerian digraphs

“…Recently, Lochet [17] proved that a digraph with high enough minimum out-degree contains an immersion of a transitive tournament. Nevertheless, there are digraphs with arbitrarily large minimum in-and out-degree which do not contain an immersion of − → K 3 (see Mader [18], who used a construction of Thomassen [20] of a family of digraphs with arbitrarily large minimum outdegree with no even directed cycle; see also [4] for a different construction).…”

Immersion of complete digraphs in Eulerian digraphs

“…Recently, Lochet [10] proved that a digraph with high enough minimum out-degree contains an immersion of a transitive tournament. Nevertheless, there are digraphs with arbitrarily large minimum in-and out-degree which do not contain an immersion of − → K 3 (see Mader [11], who used a construction of Thomassen [13] of a family of digraphs with arbitrarily large minimum out-degree with no even directed cycle; see also [3] for a different construction).…”