Subdivisions in Digraphs of Large Out-Degree or Large Dichromatic Number

Subdivisions in digraphs of large out-degree or large dichromatic number∗

In 1985, Mader conjectured the existence of a function f such that every digraph with minimum out-degree at least f(k) contains a subdivision of the transitive tournament of order k. This conjecture is still completely open, as the existence of f(5) remains unknown. In this paper, we show that if D is an oriented path, or an in-arborescence (i.e., a tree with all edges oriented towards the root...

Subdivisions of oriented cycles in digraphs with large chromatic number

A note on complete subdivisions in digraphs of large outdegree

Mader conjectured that for all ` there is an integer δ(`) such that every digraph of minimum outdegree at least δ(`) contains a subdivision of a transitive tournament of order `. In this note we observe that if the minimum outdegree of a digraph is sufficiently large compared to its order then one can even guarantee a subdivision of a large complete digraph. More precisely, let ~ G be a digraph...

Dichromatic number, circulant tournaments and Zykov sums of digraphs

The dichromatic number dc(D) of a digraph D is the smallest number of colours needed to colour the vertices of D so that no monochromatic directed cycle is created. In this paper the problem of computing the dichromatic number of a Zykov-sum of digraphs over a digraph D is reduced to that of computing a multicovering number of an hypergraph H1(D) associated to D in a natural way. This result al...

Out-degree reducing partitions of digraphs

Let $k$ be a fixed integer. We determine the complexity of finding a $p$-partition $(V_1, \dots, V_p)$ of the vertex set of a given digraph such that the maximum out-degree of each of the digraphs induced by $V_i$, ($1\leq i\leq p$) is at least $k$ smaller than the maximum out-degree of $D$. We show that this problem is polynomial-time solvable when $p\geq 2k$ and ${\cal NP}$-complete otherwise...