Observability of the extended Fibonacci cubes

A Fibonacci string of order n is a binary string of length n with no two consecutive ones. The Fibonacci cube n 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 ( i n) = V i n is de2ned recursively by V i n+2 = 0V i n+1 + 10V i n; with initial conditions V i i = Bi, V i i+1 = 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( i n) = n+ 1 when i= 1 and 2, and obtain bounds on obs( i n) 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. c © 2003 Elsevier Science B.V. All rights reserved.