Full edge-friendly index sets of complete bipartite graphs

‎‎Let G=(V,E)G=(V,E) be a simple graph‎. ‎An edge labeling f:E→{0,1}f:E→{0,1} induces a vertex labeling f+:V→\Z2f+:V→\Z2 defined by f+(v)≡∑uv∈Ef(uv)(mod2)f+(v)≡∑uv∈Ef(uv)(mod2) for each v∈Vv∈V‎, ‎where \Z2={0,1}\Z2={0,1} is the additive group of order 2‎. ‎For i∈{0,1}i∈{0,1}‎, ‎let‎ ‎ef(i)=|f−1(i)|ef(i)=|f−1(i)| and vf(i)=|(f+)−1(i)|vf(i)=|(f+)−1(i)|‎. ‎A labeling ff is called edge-friendly if‎ ‎|ef(1)−ef(0)|≤1|ef(1)−ef(0)|≤1‎. ‎If(G)=vf(1)−vf(0)If(G)=vf(1)−vf(0) is called the edge-friendly index of GG under an edge-friendly labeling ff‎. ‎The full edge-friendly index set of a graph GG is the set of all possible edge-friendly indices of GG‎. ‎Full edge-friendly index sets of complete bipartite graphs will be determined‎.