On edge-balance index sets of the complete graphs

Let G be a simple graph with vertex set ^V(G) and edge set ^E(G), and let ^Z₂^{= {0, 1}}, For a given binary edge labeling ^f:E(G)→Z₂, the edge labeling f induces a partial vertex labeling ^f*:V(G)→Z₂ such that ^{f*:V(G) = 1(0)} iff the number of 1-edges (0-edges) is strictly greater than the number of 0-edges (l-edges) incident to v, otherwise ^f*(v) is undefined. For ^iϵZ₂, let v(i) = card{vϵV(G): ^{f*(v) = i}}. and e(i) = card{eϵE(G):f(e) = i}, The edge-balance index sets of a graph ^G, ^EBI(G), is defined as ^{{|v(0)-v(1)|:} the edge labeling f satisfies ^{|e(0)-e(1)|≤1}}. In this paper, we completely determine the edge-balance index sets of the complete graphs with constructive proof.