3-Arc-Dominated Digraphs

Abstract

An oriented simple digraph $D=(V,A)$ with the minimum outdegree $d$ is called $d$-arc-dominated if for every arc $(x,y)\in A$ there is a vertex $u\in V$ with the outdegree $d$ such that both $(u,x)\in A$ and $(u,y)\in A$ hold. At the 20th British combinatorial conference, Lichiardopol posed the problem of characterizing $d$-arc-dominated digraphs. He also has posed the conjecture that a $d$-arc-dominated digraph with $d\geq2k-1$ contains $k$ vertex-disjoint directed cycles. In this paper, we give a characterization for 3-arc-dominated digraphs. Based on this characterization, we classify all 3-arc-dominated digraphs and show that the above conjecture is true when $d=3$.