Full friendly index sets of slender and flat cylinder graphs

Document Type: Research Paper

Authors

Hong Kong Baptist University

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 $P_2\times P_n$‎, ‎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 $C_m\times P_2$ for $m\geq 3$‎, ‎$C_m\times P_3$ for $m\geq 4$‎
‎and $C_3\times P_n$ for $n\geq 4$‎. ‎The results in this paper complement the existing results‎
‎in literature‎, ‎so the full friendly index set of cylinder graphs are completely determined‎.

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