Shiu, W., Ho, M. (2013). Full friendly index sets of slender and flat cylinder graphs. Transactions on Combinatorics, 2(4), 63-80. doi: 10.22108/toc.2013.3678
Wai Chee Shiu; Man-Ho Ho. "Full friendly index sets of slender and flat cylinder graphs". Transactions on Combinatorics, 2, 4, 2013, 63-80. doi: 10.22108/toc.2013.3678
Shiu, W., Ho, M. (2013). 'Full friendly index sets of slender and flat cylinder graphs', Transactions on Combinatorics, 2(4), pp. 63-80. doi: 10.22108/toc.2013.3678
Shiu, W., Ho, M. Full friendly index sets of slender and flat cylinder graphs. Transactions on Combinatorics, 2013; 2(4): 63-80. doi: 10.22108/toc.2013.3678
Full friendly index sets of slender and flat cylinder graphs
Let $G=(V,E)$ be a connected simple graph. A labeling $f:V \to Z_2$ 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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