Embedding of generalized Fibonacci cubes in hypercubes with faulty nodes

The generalized Fibonacci cubes (abbreviated to GFCs) were recently proposed as a class of interconnection topologies, which cover a spectrum ranging from regular graphs such as the hypercube to semiregular graphs such as the second order Fibonacci cube. It has been shown that the kth order GFC of dimension n+k is equivalent to an n-cube for 0⩽n<k; and it is a proper subgraph of an n-cube for n⩾k. Thus, a kth order GFC of dimension n+k can be obtained from the n-cube for all n⩾k by removing certain nodes from an n-cube. This problem is very simple when no faulty node exists in an k-cube; but it becomes very complex if some faulty nodes appear in an n-cube. In this paper, we first consider the following open problem: How can a maximal (in terms of the number of nodes) generalized Fibonacci cube be distinguished from a faulty hypercube which can also be considered as a fault-tolerant embedding in hypercubes. Then, we shall show how to directly embed a GFC into a faulty hypercube and prove that if no more than three faulty nodes exist, then an [n/2]th order GFC of dimension n+[n/2] can be directly embedded into an n-cube in the worst case, for n=4 or n⩾6