Optimal adaptive fault diagnosis of cubic Hamiltonian graphs

We study the problem of adaptive fault diagnosis for multiprocessor systems modeled by cubic Hamiltonian graphs. Each node in a system is either faulty or fault-free, and the aim of fault diagnosis is to identify correctly faulty/fault-free status of all nodes. In order to achieve it, each node tests their neighbors and output the results of tests. If the test node is fault-free, it always outputs correct test results, but if the test node is faulty, the result of the test cannot be trusted. We give a sufficient condition for a cubic Hamiltonian graph to be adoptively diagnosed in 3 testing rounds, provided that each node participates in at most one test of each round. It is the optimal number of testing rounds. The class of these graphs that satisfy this condition contains several important networks.