Digraph analogues for the Nine Dragon Tree Conjecture

Abstract

AbstractThe fractional arboricity of a digraph , denoted by , is defined as . Frank proved that a digraph decomposes into branchings, if and only if and . In this paper, we study digraph analogues for the Nine Dragon Tree Conjecture. We conjecture that, for positive integers and , if is a digraph with and , then decomposes into branchings with . This conjecture, if true, is a refinement of Frank's characterization. A series of acyclic bipartite digraphs is also presented to show the bound of given in the conjecture is best possible. We prove our conjecture for the cases . As more evidence to support our conjecture, we prove that if is a digraph with the maximum average degree and , then decomposes into pseudo‐branchings with .