Observability of a graph G is the minimum k for which the edges of G can be properly coloured with k colours in such a way that colour sets of vertices of G (sets of colours of their incident edges) are pairwise distinct. It is shown that limn--oo obs~Qn~ = 1 + q* where n q* = 0.293815... is the unique solution of the equation (x + 1)~+1 = 2x • in the interval (0, c~).
Let G be a finite undirected graph without loops and multiple edges with at most one component Kl and no component K2 (basic notions and notations can be found in [3]). For integers p,q let [p,q] be the integer interval consisting of all i with p<<. i<~q, and [p,c~) the interval of all integers >~p. For real intervals we use notations like (a,b), (a,b) and so on. For a map ¢p E [1,k] e(°~ (a k-edge-colouring of G) and a vertex x E V(G), we shall denote by Im~o(x) the colour set of x with respect to q~, i.e., the set of colours of edges incident with x. The colouring q~ is proper if limbo(y)[ is equal to the degree of y for any y E V(G); it is point-distinguishing if Im~0(y) # Im~0(z) whenever y, z E V(G), y # z. Denote by Obsk(G) the set of all proper point-distinguishing k-edge-colourings of G and by obs(G) the minimum k E [0, c~) such that Obsk(G) # 0; obs(G) is the observability of G. This graph invariant, introduced in (~ern~, et al. [1], represents a natural complement of the notions of point-distinguishing chromatic index [4], line-distinguishing chromatic number [2] and harmonious chromatic number [6]. The values of observability have been determined for complete graphs, complete bipartite graphs, paths, cycles and wheels [1] and for complete multipartite graphs with equipotent parts (see [5]).
The n-dimensional cube Q, is the graph defined recursively as follows: Q0 = K1 and Qn+l is the Cartesian product of Qn and/(2 for n E [0,c~). Thus the vertices of Qn can be represented as words of length n over the alphabet {0, 1} or n-dimensional vectors over GF(2). Vertices x, y E V(Q~) are joined by an edge just if x and y differ