Extreme edge-friendly indices of complete bipartite graphs

Let G=(V,E) be a simple graphý. ýAn edge labeling f:E→{0,1} induces a vertex labeling f :V→Z 2 defined by f (v)≡∑ uv∈E f(uv)(mod2) for each v∈V ý, ýwhere Z 2 ={0,1} is the additive group of order 2ý. ýFor i∈{0,1} ý, ýletý ýe f (i)=|f −1 (i)| and v f (i)=|(f ) −1 (i)| ý. ýA labeling f is called edge-friendly ifý ý|e f (1)−e f (0)|≤1 ý. ýI f (G)=v f (1)−v f (0) is called the edge-friendly index of G under an edge-friendly labeling f ý. ýExtreme values of edge-friendly index of complete bipartite graphs will be determinedý.