On the total detection numbers of complete bipartite graphs

Abstract

Let G be a connected graph of size at least 2 and c:E(G)→{0,1,…,k−1} an edge 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 an edge labeling c of a graph G is the sum of the labels assigned to the edges in G by c. The total detection number td(G) of G is defined by td(G)=min{val(c)}, where the minimum is taken over all detectable labelings c of G. In this paper, we investigate the total detection numbers of complete bipartite graphs.