Interesting Esoterica

$H$-supermagic labelings for firecrackers, banana trees and flowers

Article by Rachel Wulan Nirmalasari Wijaya and Andrea Semaničová-Feňovčíková and Joe Ryan and Thomas Kalinowski
  • Published in 2016
  • Added on
A simple graph $G=(V,E)$ admits an $H$-covering if every edge in $E$ is contained in a subgraph $H'=(V',E')$ of $G$ which is isomorphic to $H$. In this case we say that $G$ is $H$-supermagic if there is a bijection $f:V\cup E\to\{1,\ldots\lvert V\rvert+\lvert E\rvert\}$ such that $f(V)=\{1,\ldots,\lvert V\rvert\}$ and $\sum_{v\in V(H')}f(v)+\sum_{e\in E(H')}f(e)$ is constant over all subgraphs $H'$ of $G$ which are isomorphic to $H$. In this paper, we show that for odd $n$ and arbitrary $k$, the firecracker $F_{k,n}$ is $F_{2,n}$-supermagic, the banana tree $B_{k,n}$ is $B_{1,n}$-supermagic and the flower $F_n$ is $C_3$-supermagic.

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BibTeX entry

@article{Hsupermagiclabelingsforfirecrackersbananatreesandflowers,
	title = {{\$}H{\$}-supermagic labelings for firecrackers, banana trees and flowers},
	abstract = {A simple graph {\$}G=(V,E){\$} admits an {\$}H{\$}-covering if every edge in {\$}E{\$} is
contained in a subgraph {\$}H'=(V',E'){\$} of {\$}G{\$} which is isomorphic to {\$}H{\$}. In this
case we say that {\$}G{\$} is {\$}H{\$}-supermagic if there is a bijection {\$}f:V\cup
E\to\{\{}1,\ldots\lvert V\rvert+\lvert E\rvert\{\}}{\$} such that
{\$}f(V)=\{\{}1,\ldots,\lvert V\rvert\{\}}{\$} and {\$}\sum{\_}{\{}v\in V(H'){\}}f(v)+\sum{\_}{\{}e\in
E(H'){\}}f(e){\$} is constant over all subgraphs {\$}H'{\$} of {\$}G{\$} which are isomorphic to
{\$}H{\$}. In this paper, we show that for odd {\$}n{\$} and arbitrary {\$}k{\$}, the firecracker
{\$}F{\_}{\{}k,n{\}}{\$} is {\$}F{\_}{\{}2,n{\}}{\$}-supermagic, the banana tree {\$}B{\_}{\{}k,n{\}}{\$} is
{\$}B{\_}{\{}1,n{\}}{\$}-supermagic and the flower {\$}F{\_}n{\$} is {\$}C{\_}3{\$}-supermagic.},
	url = {http://arxiv.org/abs/1607.07911v2 http://arxiv.org/pdf/1607.07911v2},
	author = {Rachel Wulan Nirmalasari Wijaya and Andrea Semani{\v{c}}ov{\'{a}}-Fe{\v{n}}ov{\v{c}}{\'{i}}kov{\'{a}} and Joe Ryan and Thomas Kalinowski},
	comment = {},
	urldate = {2017-09-11},
	archivePrefix = {arXiv},
	eprint = {1607.07911},
	primaryClass = {cs.DM},
	year = 2016,
	collections = {attention-grabbing-titles,basically-computer-science}
}