On k-orthogonal factorizations in networks

Abstract

Let m, n, k, r and ki (1 ≤i ≤ m) are positive integers such that 1 ≤n ≤ m and k1 ≥ k2 ≥⋯≥km ≥ (r + 1)k. Let G be a graph with vertex set V(G) and edge set E(G), and H1, H2,⋯,Hr be r vertex-disjoint nk-subgraphs of G. In this article, we demonstrate that a graph G with maximum degree at most $ {\sum }_{i=1}^m {k}_i-(n-1)\mathrm{k}$ has a set $ \mathcal{F}=\{{F}_1,\cdots,{F}_n\}$ of n pairwise edge-disjoint factors of G such that Fi has maximum degree at most ki for 1 ≤ i ≤ n and $ \mathcal{F}$ is k-orthogonal to every Hj for 1 ≤ j ≤ r.