Geodesic
A Note on the Total Detection Numbers of Cycles
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 2, pp. 237-247.

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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) = minval(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.
Keywords: vertex-distinguishing coloring, detectable labeling, detection number, total detection number, Hamiltonian graph 
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Escuadro, Henry E.; Fujie, Futaba; Musick, Chad E. A Note on the Total Detection Numbers of Cycles. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 2, pp. 237-247. http://geodesic.mathdoc.fr/item/DMGT_2015_35_2_a3/

[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] 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] A.C. Burris, On graphs with irregular coloring number 2, Congr. Numer. 100 (1994) 129-140.

[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

[5] G. Chartrand, H. Escuadro, F. Okamoto and P. Zhang, Detectable colorings of graphs, Util. Math. 69 (2006) 13-32.

[6] G. Chartrand, L. Lesniak and P. Zhang, Graphs and Digraphs (CRC Press, Boca Raton, FL, 2010).

[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

[8] H. Escuadro and F. Fujie-Okamoto, The total detection numbers of graphs, J. Com- bin. Math. Combin. Comput. 81 (2012) 97-119.