Abstract
Let G be a digraph with vertex set V(G) and arc set E(G) and let g = (g −, g +) and ƒ = (ƒ −, ƒ +) be pairs of positive integer-valued functions defined on V(G) such that g −(x) ⩽ ƒ −(x) and g +(x) ⩽ ƒ +(x) for each x ∈ V(G). A (g, ƒ)-factor of G is a spanning subdigraph H of G such that g −(x) ⩽ id H (x) ⩽ ƒ −(x) and g +(x) ⩽ od H (x) ⩽ ƒ +(x) for each x ∈ V(H); a (g, ƒ)-factorization of G is a partition of E(G) into arc-disjoint (g, ƒ)-factors. Let
= {F 1, F 2,…, F m} and H be a factorization and a subdigraph of G, respectively.
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,mƒ−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.
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Liu, G. Orthogonal factorizations of digraphs. Front. Math. China 4, 311–323 (2009). https://doi.org/10.1007/s11464-009-0011-y
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DOI: https://doi.org/10.1007/s11464-009-0011-y