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 Z 2 = {0, 1} , For a given binary edge labeling f:E(G)→Z 2 , the edge labeling f induces a partial vertex labeling f*:V(G)→Z 2 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 iϵZ 2 , 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.