Distributed algorithms for shortest-path, deadlock-free routing and broadcasting in a class of interconnection topologies

The (non-)existence of perfect codes in Lucas cubes

“…The most complete generalization proposed in [9] is, for a given string f , to consider Γ n (f ) the subgraph of Q n induced by strings that do not contain f as substring. Since Fibonacci cubes are Γ n (11) the most immediate generalization [15,19] is to consider Γ n (1 s ) for a given integer s. In [16] we proved the existence of a perfect code in Γ n (1 s ) for n = 2 p − 1 and s ≥ 3.2 p−2 for any integer p ≥ 2.…”

Generalized fibonacci cubes are mostly hamiltonian

“…In [3], it is considered as one of the fundamental problems in graph theory. In this work, we will study the Hamiltonian problem for the family of the generalized Fibonacci cubes [9]. Our main results are as follows: (1) each member in the family contains a Hamiltonian path, (2) each member contains a Hamiltonian cycle if and only if it has an even number of nodes, and (3) for members that do not contain Hamiltonian cycles, the longest cycles contain all but one node.…”