Recognizable colorings of graphs

Let G be a connected graph and let c : V (G) → {1, 2, . . . , k} be a coloring of the vertices of G for some positive integer k (where adjacent vertices may be colored the same). The color code of a vertex v of G (with respect to c) is the ordered (k+1)-tuple code(v) = (a0, a1, . . . , ak) where a0 is the color assigned to v and for 1 ≤ i ≤ k, ai is the number of vertices adjacent to v that are colored i. The coloring c is called recognizable if distinct vertices have distinct color codes and the recognition number rn(G) of G is the minimum positive integer k for which G has 36 G. Chartrand, L. Lesniak, D.W. VanderJagt and P. Zhang a recognizable k-coloring. Recognition numbers of complete multipartite graphs are determined and characterizations of connected graphs of order n having recognition numbers n or n− 1 are established. It is shown that for each pair k, n of integers with 2 ≤ k ≤ n, there exists a connected graph of order n having recognition number k. Recognition numbers of cycles, paths, and trees are investigated.