Full edge-friendly index sets of complete bipartite graphs

‎‎Let G=(V,E) be a simple graph‎. ‎An edge labeling f:E\to \{۰,۱\} induces a vertex labeling f^+:V\to\Z_۲ defined by f^+(v)\equiv \sum\limits_{uv\in E} f(uv)\pmod{۲} for each v \in V‎, ‎where \Z_۲=\{۰,۱\} is the additive group of order ۲‎. ‎For i\in\{۰,۱\}‎, ‎let‎ ‎e_f(i)=|f^{-۱}(i)| and v_f(i)=|(f^+)^{-۱}(i)|‎. ‎A labeling f is called edge-friendly if‎ ‎|e_f(۱)-e_f(۰)|\le ۱‎. ‎I_f(G)=v_f(۱)-v_f(۰) is called the edge-friendly index of G under an edge-friendly labeling f‎. ‎The full edge-friendly index set of a graph G is the set of all possible edge-friendly indices of G‎. ‎Full edge-friendly index sets of complete bipartite graphs will be determined‎.

Keywords:

‎Full edge-friendly index sets‎ , ‎edge-friendly index‎ , ‎edge-friendly labeling‎ , ‎complete bipartite graph