Observability of the extended Fibonacci cubes Original Research Article

A Fibonacci string of order n is a binary string of length n with no two consecutive ones. The Fibonacci cube Γn is the subgraph of the hypercube Qn induced by the set of Fibonacci strings of order n. For positive integers i,n, with n⩾i, the ith extended Fibonacci cube is the vertex induced subgraph of Qn for which V(Γni)=Vni is defined recursively byVn+2i=0Vn+1i+10Vni,with initial conditions Vii=Bi, Vi+1i=Bi+1, where Bk denotes the set of binary strings of length k. A proper edge colouring of a simple graph G is called strong if it is vertex distinguishing. The observability of G, denoted by obs(G), is the minimum number of colours required for a strong edge colouring of G. In this study we prove that obs(Γni)=n+1 when i=1 and 2, and obtain bounds on obs(Γni) for i⩾3 which are sharp in some cases. We also obtain bounds on the value of obs(G×Qn), n⩾2, for a graph G containing at most one isolated vertex and no isolated edge.