Randomly orthogonal factorizations with constraints in bipartite networks

Many problems on computer science, chemistry, physics and network theory are related to factors, factorizations and orthogonal factorizations in graphs. For example, the telephone network design problems can be converted into maximum matchings of graphs; perfect matchings or 1-factors in graphs correspond to Kekulé structures in chemistry; the file transfer problems in computer networks can be modelled as (0, f)-factorizations in graphs; the designs of Latin squares and Room squares are related to orthogonal factorizations in graphs; the orthogonal (g, f)-colorings of graphs are related to orthogonal (g, f)-factorizations of graphs. In this paper, the orthogonal factorizations in graphs are discussed and we show that every bipartite (0,mf−(m−1)r)-graph G has a (0, f)-factorization randomly r-orthogonal to n vertex disjoint mr-subgraphs of G in certain conditions.