Extreme edge-friendly indices 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. ‎Extreme values of edge-friendly index of complete bipartite graphs will be determined‎.

کلیدواژه ها:

‎edge-friendly index‎ ، ‎edge-friendly labeling‎ ، ‎complete bipartite graph