On Edge-Balance Index Sets of Regular Graphs

In a bi-racial country, it is desired that the numbers of government ministers from the two races should differ by at most one for fairness. Moreover, the numbers of pairs of ministries which interact directly with each other and both headed by ministers of one race should differ by at most one from that of the other race. One can naturally model the above social phenomenon by a graph labeling which is called edge-balanced, which assigns edges approximately half 0’s, and the other half 1’s. And then the labeling requires that the induced vertex labels are also 0’s over approximately one half vertices, and 1’s over the other one half vertices. We generalize the concept of edge-balanced labeling to the concept of edge-balance index sets of graphs. In this article, properties regarding the edge-balance index sets of regular graphs are investigated. In particular, we completely determine the edge-balance index sets of all 2-regular graphs, Möbius ladders (as examples of 3-regular graphs), and moreover, complete graphs (as examples of general regular graphs) among others.