On the total detection numbers of complete bipartite graphs

Let GG be a connected graph of size at least 2 and c:E(G)→{0,1,…,k−1}c:E(G)→{0,1,…,k−1} an edge labeling of GG using kk labels, where adjacent edges may be assigned the same label. For each vertex vv of GG, the color code of vv with respect to cc is the kk-vector code(v)=(a0,a1,…,ak−1), where aiai is the number of edges incident with vv that are labeled ii for 0≤i≤k−10≤i≤k−1. The labeling cc is called a detectable labeling if distinct vertices in GG have distinct color codes. The value val(c) of an edge labeling cc of a graph GG is the sum of the labels assigned to the edges in GG by cc. The total detection number td(G) of GG is defined by td(G)=min{val(c)}, where the minimum is taken over all detectable labelings cc of GG. In this paper, we investigate the total detection numbers of complete bipartite graphs.