Geodesic
Detection number of bipartite graphs and cubic graphs
Discrete mathematics & theoretical computer science, Tome 16 (2014) no. 3.

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For a connected graph G of order |V(G)| ≥3 and a k-labelling c : E(G) →{1,2,…,k} of the edges of G, the code of a vertex v of G is the ordered k-tuple (ℓ1,ℓ2,…,ℓk), where ℓi is the number of edges incident with v that are labelled i. The k-labelling c is detectable if every two adjacent vertices of G have distinct codes. The minimum positive integer k for which G has a detectable k-labelling is the detection number det(G) of G. In this paper, we show that it is NP-complete to decide if the detection number of a cubic graph is 2. We also show that the detection number of every bipartite graph of minimum degree at least 3 is at most 2. Finally, we give some sufficient condition for a cubic graph to have detection number 3. 
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     author = {Havet, Frederic and Paramaguru, Nagarajan and Sampathkumar, Rathinaswamy},
     title = {Detection number of bipartite graphs and cubic graphs},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {16},
     number = {3},
     year = {2014},
     doi = {10.46298/dmtcs.642},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.642/}
}
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Havet, Frederic; Paramaguru, Nagarajan; Sampathkumar, Rathinaswamy. Detection number of bipartite graphs and cubic graphs. Discrete mathematics & theoretical computer science, Tome 16 (2014) no. 3. doi : 10.46298/dmtcs.642. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.642/

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