On Additive Bases and Harmonious Graphs

This paper first considers several types of additive bases. A typical problem is to find nv(k), the largest n for which there exists a set {0 al < a2 <" < ak} Of distinct integers modulo n such that each in the range 0 =<-< n can be written at least once as mai + aj (modulo n) with </'. For example, nv(8) 24, The other problems arise if at least is changed to at most, or </' to-</', or if the words modulo n are omitted. Tables and bounds are given for each of these problems. Then a closely related graph labeling problem is studied. A connected graph with n edges is called harmonious if it is possible to label the vertices with distinct numbers (modulo n) in such a way that the edge sums are also distinct (modulo n). Some infinite families of graphs (odd cycles, ladders, wheels,...) are shown to be harmonious while others (even cycles, most complete or complete bipartite graphs, .) are not. In fact most graphs are not harmonious. The function nv(k) is the size of the largest harmonious subgraph of the complete graph on k vertices.