Orthogonal factorizations of digraphs

Abstract

Let G be a digraph with vertex set V (G) and arc set E(G) and let g = (g^-, g⁺) and f = (f^-, f⁺) be pairs of positive integer-valued functions defined on V (G) such that g^-(x)≤f^-(x) and g⁺(x)≤f⁺(x) for each x ∈ V (G). A (g, f)-factor of G is a spanning subdigraph H of G such that g^-(x)≤id_H(x)≤f^-(x) and g⁺(x)≤od_H(x)≤f⁺(x) for each x ∈ V (H); a (g, f)-factorization of G is a partition of E(G) into arc-disjoint (g, f)-factors. Let ℱ=\left\{F_1,F_2,\cdots ,F_m\right\} and H be a factorization and a subdigraph of G, respectively. ℱ is called k-orthogonal to H if each F_i, 1≤i≤m, has exactly k arcs in common with H. In this paper it is proved that every (mg+m-1,mf-m+1)-digraph has a (g, f)-factorization k-orthogonal to any given subdigraph with km arcs if k≤min{g^-(x), g⁺(x)} for any x ∈ V (G) and that every (mg,mf)-digraph has a (g, f)-factorization orthogonal to any given directed m-star if 0≤g(x)≤f(x) for any x ∈ V (G). The results in this paper are in some sense best possible.

Keywords

Digraph / (g, f)-factor / orthogonal factorization

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