One-to-One Disjoint Path Covers in Dragonfly Networks

Dragonfly networks are scalable interconnection topologies for large-scale high-performance computing systems. Given two distinct vertices \(u\) and \(v\) of a graph, a one-to-one \(m\)-disjoint path cover (\(m\)-DPC) is a family of \(m\) internally vertex-disjoint \(u\text{--}v\) paths whose union contains every vertex. We determine the complete feasible range of \(m\) for the dragonfly network \(D(n,h)\). After choosing one group as a core, we partition the remaining groups into \(h\) sets, each inducing a two-level Sierpiński graph \(S(2,n)\), and exploit Hamilton paths, spanning fans, and the matching structure among these induced subgraphs to construct the required covers. For \(n\ge4\) and \(h\ge2\), \(D(n,h)\) admits a one-to-one \(m\)-DPC for every integer \(m\) satisfying \(1\le m\le n+h-1\). This range is optimal because \(D(n,h)\) is \((n+h-1)\)-regular.