Full friendly index sets of slender and flat cylinder graphs

Let $G=(V,E)$ be a connected simple graph. A labeling $f:Vrightarrow Z_2$ induces an edgelabeling $f^*:EtoZ_2$ defined by $f^*(xy)=f(x)+f(y)$ for each $xy in E$. For $iinZ_2$,let $v_f(i)=|f^{-1}(i)|$ and $e_f(i)=|f^{*-1}(i)|$. A labeling $f$ is called friendly if$|v_f(1)-v_f(0)|le 1$. The full friendly index set of $G$ consists all possible differencesbetween the number of edges labeled by 1 and the number of edges labeled by 0. In recent years,full friendly index sets for certain graphs were studied, such as tori, grids $P_2times P_n$,and cylinders $C_mtimes P_n$ for some $n$ and $m$. In this paper we study the full friendlyindex sets of cylinder graphs $C_mtimes P_2$ for $mgeq 3$, $C_mtimes P_3$ for $mgeq 4$and $C_3times P_n$ for $ngeq 4$. The results in this paper complement the existing resultsin literature, so the full friendly index set of cylinder graphs are completely determined.