The overall performance of Network-on-Chip (NoC) is strongly affected by the efficiency of the on-chip routing algorithm. Among the factors associated with the design of a high-performance routing method, adaptivity is an important one. Moreover, deadlock- and livelock-freedom are necessary for a functional routing method. Despite the advantages that the diametrical mesh can bring to NoCs compared with the classical mesh topology, the literature records little research efforts to design pertinent routing methods for such networks. Using the available routing algorithms, the network performance degrades drastically not only due to the deterministic paths, but also to the deadlocks created between the packets. In this paper, we take advantage of the Hamiltonian routing strategy to adaptively route the packets through deadlock-free paths in a diametrical 2D mesh network. The simulation results demonstrate the efficiency of the proposed approach in decreasing the likelihood of congestion and smoothly distributing the traffic across the network.