Geodesic
Recognizable colorings of graphs
Discussiones Mathematicae. Graph Theory, Tome 28 (2008) no. 1, pp. 35-57

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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, 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.
Keywords: recognizable coloring, recognition number 
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Chartrand, Gary; Lesniak, Linda; VanderJagt, Donald; Zhang, Ping. Recognizable colorings of graphs. Discussiones Mathematicae. Graph Theory, Tome 28 (2008) no. 1, pp. 35-57. http://geodesic.mathdoc.fr/item/DMGT_2008_28_1_a2/