# 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.

Keywords

Main Subjects

### References

J. A. Bondy and U. S. R. Murty (2008). Graph Theory. Graduate Texts in Mathematics, Springer, New York. 244
G. Chartrand, S.-M. Lee and P. Zhang (2006). Uniformly cordial graphs. Discrete Math.. 306, 726-737
H. Kwong, S.-M. Lee and H. K. Ng (2008). On friendly index sets of $2$-regular graphs. Discrete Math.. 308, 5522-5532
H. Kwong and S.-M. Lee (2008). On friendly index sets of generalized books. J. Combin. Math. Combin. Comput.. 66, 43-58
S.-M. Lee and H. K. Ng (2008). On friendly index sets of bipartite graphs. Ars Combin.. 86, 257-271
E. Salehi and S.-M. Lee (2006). On friendly index sets of trees. Congr. Numer.. 178, 173-183
W. C. Shiu and H. Kwong (2008). Full friendly index sets of {P}_2times{P}_n. Discrete Math.. 308, 3688-3693
W. C. Shiu and M. H. Ling (2007). Extreme friendly indices of {$C_mtimes C_n$}. Congr. Numer.. 188, 175-182
W. C. Shiu and M. H. Ling (2010). Full friendly index sets of {C}artesian products of two cycles. Acta Math. Sin. (Engl. Ser.). 26, 1233-1244
W. C. Shiu and F. S. Wong (2009). W. C. Shiu and F. S. Wong. Congr. Numer.. 197, 65-75
W. C. Shiu and F. S. Wong (2012). Full friendly index sets of cylinder graphs. Australas. J. Combin.. 52, 141-162