# On partitioning the edges of an infinite digraph into directed cycles

## TL;DR

This paper proves a conjecture that the edges of an infinite directed graph can be partitioned into directed cycles if and only if each subset of vertices has equal ingoing and outgoing edges, extending a classical undirected graph result.

## Contribution

It provides a proof for Thomassen's conjecture on partitioning edges of infinite digraphs into directed cycles based on vertex in-out degree conditions.

## Key findings

- Confirmed the conjecture for infinite digraphs.
- Established necessary and sufficient conditions for cycle partitioning.
- Extended classical undirected graph results to directed infinite graphs.

## Abstract

Nash-Williams proved that for an undirected graph $ G $ the set $ E(G) $ can be partitioned into cycles if and only if every cut has either even or infinite number of edges. Later C. Thomassen gave a simpler proof for this and conjectured the following directed analogue of the theorem: the edge set of a digraph can be partitioned into directed cycles if and only if for each subset of the vertices the cardinality of the ingoing and the outgoing edges are equal. The aim of the paper is to prove this conjecture.