Recognizable colorings of cycles and trees

For a graph G and a vertex-coloring c : V (G) → {1, 2, . . . , k}, the color code of a vertex v is the (k + 1)-tuple (a0, a1, . . . , ak), where a0 = c(v), and for 1 ≤ i ≤ k, ai is the number of neighbors of v colored i. A recognizable coloring is a coloring such that distinct vertices have distinct color codes. The recognition number of a graph is the minimum k for which G has a recognizable k-coloring. In this paper we prove three conjectures of Chartrand et al. in [8] regarding the recognition number of cycles and trees.