Detection number of bipartite graphs and cubic graphs

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.