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, ${\displaystyle a_i}$ 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 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.