Recognizable colorings of graphs
Gary Chartrand, Linda M. Lesniak, Donald W. VanderJagt, Ping Zhang · Discussiones Mathematicae Graph Theory · 2008
Let G be a connected graph and let c : V (G) ! f1; 2; : : : ; kg 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