Geodesic
Recognizable colorings of cycles and trees
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 1, pp. 81-90

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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, a_i 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.
Keywords: recognizable coloring, recognition number 
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Dorfling, Michael; Dorfling, Samantha. Recognizable colorings of cycles and trees. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 1, pp. 81-90. http://geodesic.mathdoc.fr/item/DMGT_2012_32_1_a6/