# Recognizable colorings of graphs

• Volume: 28, Issue: 1, page 35-57
• ISSN: 2083-5892

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## Abstract

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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.

## How to cite

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Gary Chartrand, et al. "Recognizable colorings of graphs." Discussiones Mathematicae Graph Theory 28.1 (2008): 35-57. <http://eudml.org/doc/270473>.

@article{GaryChartrand2008,
abstract = {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.},
author = {Gary Chartrand, Linda Lesniak, Donald W. VanderJagt, Ping Zhang},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {recognizable coloring; recognition number},
language = {eng},
number = {1},
pages = {35-57},
title = {Recognizable colorings of graphs},
url = {http://eudml.org/doc/270473},
volume = {28},
year = {2008},
}

TY - JOUR
AU - Gary Chartrand
AU - Linda Lesniak
AU - Donald W. VanderJagt
AU - Ping Zhang
TI - Recognizable colorings of graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2008
VL - 28
IS - 1
SP - 35
EP - 57
AB - 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.
LA - eng
KW - recognizable coloring; recognition number
UR - http://eudml.org/doc/270473
ER -

## References

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10. [10] H. Escuadro, F. Okamoto and P. Zhang, A three-color problem in graph theory, Bull. Inst. Combin. Appl., to appear. Zbl1146.05022
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