Arc-disjoint strong spanning subdigraphs in compositions and products of digraphs

Abstract

A digraph D=(V,A) has a good decomposition if A has two disjoint sets A1 and A2 such that both (V,A1) and (V,A2) are strong. Let T be a digraph with vertices u1,…,ut (t≥2) and let H1,…Ht be digraphs such that Hi has vertices ui,ji,1≤ji≤ni. Then the compositionQ=T[H1,…,Ht] is a digraph with vertex set {ui,ji:1≤i≤t,1≤ji≤ni} and arc set A(Q)=∪i=1tA(Hi)∪{uijiupqp:uiup∈A(T),1≤ji≤ni,1≤qp≤np}. For digraph compositions Q=T[H1,…Ht], we obtain sufficient conditions for Q to have a good decomposition and a characterization of Q with a good decomposition when T is a strong semicomplete digraph and each Hi is an arbitrary digraph with at least two vertices.For digraph products, we prove the following: (a) if k≥2 is an integer and G is a strong digraph which has a collection of arc-disjoint cycles covering all vertices, then the Cartesian product digraph G□k (the kth power of G with respect to Cartesian product) has a good decomposition; (b) for any strong digraphs G,H, the strong product G⊠H has a good decomposition.