Full friendly index sets of slender and flat cylinder graphs

‎Let G=(V,E) be a connected simple graph‎. ‎A labeling f:V \to Z_۲ induces an edge labeling‎ ‎f^*:E \to Z_۲ defined by f^*(xy)=f(x)+f(y) for each xy \in E‎. ‎For i \in Z_۲‎, ‎let‎ ‎v_f(i)=|f^{-۱}(i)| and e_f(i)=|f^{*-۱}(i)|‎. ‎A labeling f is called friendly if‎ ‎|v_f(۱)-v_f(۰)|\le ۱‎. ‎The full friendly index set of  G consists all possible differences‎ ‎between the number of edges labeled by ۱ and the number of edges labeled by ۰‎. ‎In recent years‎, ‎full friendly index sets for certain graphs were studied‎, ‎such as tori‎, ‎grids P_۲\times P_n‎, ‎and cylinders C_m\times P_n for some n and m‎. ‎In this paper we study the full friendly‎ ‎index sets of cylinder graphs C_m\times P_۲ for m\geq ۳‎, ‎C_m\times P_۳ for m\geq ۴‎ ‎and C_۳\times P_n for n\geq ۴‎. ‎The results in this paper complement the existing results‎ ‎in literature‎, ‎so the full friendly index set of cylinder graphs are completely determined‎.

Keywords:

Full friendly index sets , friendly labeling , cylinder graphs