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, 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1390,
author = {Gary Chartrand and Linda Lesniak and Donald W. VanderJagt and Ping Zhang},
title = {Recognizable colorings of graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {28},
year = {2008},
pages = {35-57},
zbl = {1235.05049},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1390}
}
Gary Chartrand; Linda Lesniak; Donald W. VanderJagt; Ping Zhang. Recognizable colorings of graphs. Discussiones Mathematicae Graph Theory, Tome 28 (2008) pp. 35-57. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1390/
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