# A Note on the Total Detection Numbers of Cycles

• Volume: 35, Issue: 2, page 237-247
• ISSN: 2083-5892

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## Abstract

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Let G be a connected graph of size at least 2 and c :E(G)→{0, 1, . . . , k− 1} an edge coloring (or labeling) of G using k labels, where adjacent edges may be assigned the same label. For each vertex v of G, the color code of v with respect to c is the k-vector code(v) = (a0, a1, . . . , ak−1), where ai is the number of edges incident with v that are labeled i for 0 ≤ i ≤ k − 1. The labeling c is called a detectable labeling if distinct vertices in G have distinct color codes. The value val(c) of a detectable labeling c of a graph G is the sum of the labels assigned to the edges in G. The total detection number td(G) of G is defined by td(G) = min{val(c)}, where the minimum is taken over all detectable labelings c of G. We investigate the problem of determining the total detection numbers of cycles.

## How to cite

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Henry E. Escuadro, Futaba Fujie, and Chad E. Musick. "A Note on the Total Detection Numbers of Cycles." Discussiones Mathematicae Graph Theory 35.2 (2015): 237-247. <http://eudml.org/doc/271101>.

@article{HenryE2015,
abstract = {Let G be a connected graph of size at least 2 and c :E(G)→\{0, 1, . . . , k− 1\} an edge coloring (or labeling) of G using k labels, where adjacent edges may be assigned the same label. For each vertex v of G, the color code of v with respect to c is the k-vector code(v) = (a0, a1, . . . , ak−1), where ai is the number of edges incident with v that are labeled i for 0 ≤ i ≤ k − 1. The labeling c is called a detectable labeling if distinct vertices in G have distinct color codes. The value val(c) of a detectable labeling c of a graph G is the sum of the labels assigned to the edges in G. The total detection number td(G) of G is defined by td(G) = min\{val(c)\}, where the minimum is taken over all detectable labelings c of G. We investigate the problem of determining the total detection numbers of cycles.},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {vertex-distinguishing coloring; detectable labeling; detection number; total detection number; Hamiltonian graph},
language = {eng},
number = {2},
pages = {237-247},
title = {A Note on the Total Detection Numbers of Cycles},
url = {http://eudml.org/doc/271101},
volume = {35},
year = {2015},
}

TY - JOUR
AU - Futaba Fujie
TI - A Note on the Total Detection Numbers of Cycles
JO - Discussiones Mathematicae Graph Theory
PY - 2015
VL - 35
IS - 2
SP - 237
EP - 247
AB - Let G be a connected graph of size at least 2 and c :E(G)→{0, 1, . . . , k− 1} an edge coloring (or labeling) of G using k labels, where adjacent edges may be assigned the same label. For each vertex v of G, the color code of v with respect to c is the k-vector code(v) = (a0, a1, . . . , ak−1), where ai is the number of edges incident with v that are labeled i for 0 ≤ i ≤ k − 1. The labeling c is called a detectable labeling if distinct vertices in G have distinct color codes. The value val(c) of a detectable labeling c of a graph G is the sum of the labels assigned to the edges in G. The total detection number td(G) of G is defined by td(G) = min{val(c)}, where the minimum is taken over all detectable labelings c of G. We investigate the problem of determining the total detection numbers of cycles.
LA - eng
KW - vertex-distinguishing coloring; detectable labeling; detection number; total detection number; Hamiltonian graph
UR - http://eudml.org/doc/271101
ER -

## References

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1. [1] M. Aigner and E. Triesch, Irregular assignments and two problems à la Ringel , in: Topics in Combinatorics and Graph Theory, R. Bodendiek and R. Henn (Ed(s)), (Heidelberg: Physica, 1990) 29-36.
2. [2] M. Aigner, E. Triesch and Zs. Tuza, Irregular assignments and vertex-distinguishing edge-colorings of graphs, in: Combinatorics ’90, (New York: Elsevier Science Pub., 1992) 1-9.
3. [3] A.C. Burris, On graphs with irregular coloring number 2, Congr. Numer. 100 (1994) 129-140. Zbl0836.05029
4. [4] A.C. Burris, The irregular coloring number of a tree, Discrete Math. 141 (1995) 279-283. doi:10.1016/0012-365X(93)E0225-S[Crossref] Zbl0829.05027
5. [5] G. Chartrand, H. Escuadro, F. Okamoto and P. Zhang, Detectable colorings of graphs, Util. Math. 69 (2006) 13-32. Zbl1102.05020
6. [6] G. Chartrand, L. Lesniak and P. Zhang, Graphs and Digraphs (CRC Press, Boca Raton, FL, 2010). Zbl1329.05001
7. [7] H. Escuadro, F. Fujie and C.E. Musick, On the total detection numbers of complete bipartite graphs, Discrete Math. 313 (2013) 2908-2917. doi:10.1016/j.disc.2013.09.001[Crossref] Zbl1281.05120
8. [8] H. Escuadro and F. Fujie-Okamoto, The total detection numbers of graphs, J. Com- bin. Math. Combin. Comput. 81 (2012) 97-119. Zbl1252.05060

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