A generalization of orthogonal factorizations in digraphs

Orthogonal factorizations in graphs or networks have a wide range of applications in combinatorial design, network design, circuit layout, and so on. We investigate the problem on orthogonal factorizations in graphs or networks. Let G be a digraph. Its vertex set and arc set are denoted by V(G) and E(G), respectively. Let f=(f−,f+) be a pair of nonnegative integer-valued functions defined on V(G). Let H1,H2,⋯,Hr be r vertex disjoint mk-subdigraphs of G. In this paper, it is verified that every (0,mf−m+1)-digraph admits a (0,f)-factorization k-orthogonal to each Hi (i=1,2,⋯,r) if f(x)≥(2r+1)k−2 for any x∈V(G), which is a generalization of Zhou's previous result.