Two-Disjoint-Cycle-Cover Pancyclicity of Dragonfly Networks

Interconnection networks, often modeled as graphs, are critical for high-performance computing systems due to their impact on performance metrics like latency and bandwidth. The dragonfly network, denoted as \( D(n,r) \), is a promising topology owing to its modularity, low diameter, and cost-effectiveness. Ensuring reliability and efficiency in these networks requires robust cycle embedding properties. The two-disjoint-cycle-cover pancyclicity ensures that the network can be partitioned into two vertex-disjoint cycles of any feasible length, which has practical implications for fault-tolerant routing and load balancing. Formally, a graph \( G \) is called two-disjoint-cycle-cover \( [a_1,a_2] \)-pancyclic if for any integer \( \) satisfying \( a_1≤ ≤ a_2 \), there exist two vertex-disjoint cycles \( C_1 \) and \( C_2 \) in \( G \) such that \( |V(C_1)|= and |V(C_2)|=|V(G)|- \). While prior work has established Hamiltonicity and pancyclicity for \( D(n,r) \), the two-disjoint-cycle-cover problem remains unexplored. This paper fills this gap by proving that \( D(n,r) \) is two-disjoint-cycle-cover \( [3, |V(D(n,r))|/2 ] \)-pancyclic with \( n≥ 3 \) and \( r≥ 2 \), generalizing existing knowledge. Moreover, it can be obtained that \( D(n,r) \) is vertex-disjoint-cycle-cover. Our proof employs a constructive method with case analysis, ensuring the existence of such cycles.