\((g,f)\)-factorizations of graphs orthogonal to \([1,2]\)-subgraphs (Q1375339)

A graph \(G\) is called a \((g,f)\)-graph if for every vertex \(v\) of \(G\), \(g(v)\leq d_G(v)\leq f(v)\), where \(g\) and \(f\) are integer-valued functions defined on the vertex set of \(G\), \(d_G(v)\) denotes the degree of the vertex \(v\) in \(G\), and \(g(v)\leq f(v)\) for all \(v\) in \(G\). A \((g,f)\)-factorization \({\mathcal F}=\{F_1,F_2,\ldots, F_t\}\) of \(G\) is a partition of the edge set of \(G\) into spanning \((g,f)\)-subgraphs. A subgraph \(H\) of \(G\) is orthogonal to \({\mathcal F}\) if \(|E(H)\cap E(F_i)|=1\) for \(1\leq i\leq t\). Let \(m\) be a positive integer, and let \(g(v)\leq f(v)\) be integer-valued functions defined on the vertex set of \(G\). The author proves that if \(G\) is a \((mg+m-1,mf-m+1)\)-graph and \(H\) is a \([1,2]\)-subgraph with \(m\) edges, then there exists a \((g,f)\)-factorization of \(G\) orthogonal to \(H\).