# Node disjoint circuits in Eulerian digraphs

From Egres Open

Let *G* be a directed graph without oneway paralel arcs, in which every node has indegree and outdegree *k*. Is it true that there is a node *v* such that there are *k* directed cycles containing *v*, pairwise node-disjoint (except for *v*)?

Recently Wolfgang Mader ^{[1]} showed that the answer to the question is negative for any [math]k \geq 8[/math]. He also proved that the answer is positive for *k=3*, and for any *k* in vertex-transitive digraphs.

## Remarks

This question was asked by Paul Seymour. Note that it would imply that if *G* is a digraph with at most *kn* nodes and every node has indegree and outdegree *k*, then there is a directed cycle of length at most *n*, which is a very interesting open special case of the Caccetta-Häggkvist conjecture.