Full friendly index sets of slender and flat cylinder graphs

Abstract

‎Let $G=(V,E)$ be a connected simple graph‎. ‎A labeling $f:V to Z_2$ induces an edge labeling‎ ‎$f^*:E to Z_2$ defined by $f^*(xy)=f(x)+f(y)$ for each $xy in E$‎. ‎For $i in Z_2$‎, ‎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 differences‎ ‎between 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 friendly‎ ‎index sets of cylinder graphs $C_mtimes P_2$ for $mgeq 3$‎, ‎$C_mtimes P_3$ for $mgeq 4$‎ <br />‎and $C_3times P_n$ for $ngeq 4$‎. ‎The results in this paper complement the existing results‎ <br />‎in literature‎, ‎so the full friendly index set of cylinder graphs are completely determined‎.