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/