#
On Forbidden Subgraphs of (K_{2}, H)-Sim-(Super)Magic Graphs

^{*}

## Abstract

**:**

## 1. Introduction

**Lemma**

**1**

**([4]).**A graph G is SEMT if and only if there exists a bijective function $f:V\left(G\right)\to [1,|V\left(G\right)\left|\right]$ such that the set $S=\left\{f\right(u)+f(v\left)\right|uv\in E\left(G\right)\}$ consists of $\left|E\right(G\left)\right|$ consecutive integers. In such a case, f extends to an SEMT labeling of G with magic sum $k=\left|V\right(G\left)\right|+\left|E\right(G\left)\right|+s$, where $s=min\left(S\right)$.

## 2. Previous Results on H-(Super)Magic Labelings

**Theorem**

**2**

**([14]).**Let $n\ge 4$ be a positive integer.

- 1 .
- If G is ${P}_{n}$-magic, then G is ${C}_{n-1}^{+1}$-free.
- 2 .
- If G is ${P}_{n}$-magic, then G is ${C}_{n+1}^{+1}$-free.

**Theorem**

**3**

**Theorem**

**4**

**([12]).**Let f be a ${S}_{h}$-magic labeling of a graph G with magic constant ${m}_{f}$. If the degree of vertex $x\in V\left(G\right)$ verifies $deg\left(x\right)>h$, then for every vertex y adjacent to x, we have $f\left(y\right)+f\left(xy\right)=\frac{1}{h}({m}_{f}-f\left(x\right))$.

**Corollary**

**1**

**([12]).**Let G be a ${S}_{h}$-magic graph with $h>1$. Then, for every edge $e=xy$ of G, $min\left\{deg\right(x),deg(y\left)\right\}\le h$.

**Theorem**

**6**

**([16]).**For any two integers $k\ge 2$ and $r\ge 3$, the windmill $W(r,k)$ is ${C}_{r}$-supermagic.

**Theorem**

**7**

**([18]).**Let $n\ge 4$ be a positive integer.

- 1 .
- The fan ${F}_{n}$ is ${C}_{m}$-supermagic for any integer $4\le m\le \lfloor \frac{n+4}{2}\rfloor $;
- 2 .
- The ladder ${L}_{n}$ is ${C}_{m}$-supermagic for any positive integer $3\le m\le \lfloor \frac{n}{2}\rfloor +1$.

**Observation**

**1.**

## 3. $({\mathit{K}}_{\mathbf{2}},{\mathit{P}}_{\mathit{n}})$-Sim-Supermagic Labelings

**Theorem**

**8.**

- 1 .
- $H\cong {C}_{m}$, for any $n\ge 4$ and $m\in [n-1,n+1]$;
- 2 .
- $H\cong {H}_{n+2}$, for any $n\ge 3$;
- 3 .
- $H\cong {P}_{n+1}$, for any $n\ge 3$;
- 4 .
- $H\cong S({S}_{3};{e}_{1},{e}_{2},{e}_{3};n-2,3,3)$, for any $n\ge 5$;
- 5 .
- $H\cong (k,n-k)$-tadpole, for any $n>4$ and $k\in [3,n-2]$;
- 6 .
- $H\cong Amal\left(\right(m,3)$-tadpole, ${P}_{n-m};\mathcal{H};2)$, for any $n\ge 7,$ and $m\in [3,n-4]$.

**Proof.**

**Example**

**1.**

**Proof.**

**Problem**

**1.**

**Corollary**

**2.**

**Corollary**

**3.**

**Corollary**

**4.**

**Corollary**

**5.**

**Corollary**

**6.**

**Problem**

**2.**

**Lemma**

**3.**

**Proof.**

**Corollary**

**7.**

**Proof.**

## 4. A $({\mathit{K}}_{\mathbf{2}},{\mathit{S}}_{\mathit{n}})$-Sim-Supermagic Labelings

**Lemma**

**4.**

**Proof.**

**Proof.**

**Theorem**

**10.**

**Proof.**

## 5. A $({\mathit{K}}_{\mathbf{2}},{\mathit{C}}_{\mathit{n}})$-Sim-Supermagic Labelings

**Lemma**

**5.**

**Proof.**

**Corollary**

**8.**

**Corollary**

**9.**

**Proof.**

**Corollary**

**10.**

- 1 .
- ${K}_{n,n}$ is $({K}_{2},{C}_{2n})$-sim-magic;
- 2 .
- ${W}_{n}$ is $({K}_{2},{C}_{n+1})$-sim-magic for $n\equiv 0,1,$ or 2(mod 4).

**Theorem**

**11.**

**Proof.**

- 1 .
- $f\left({x}_{i}\right)+f\left({x}_{i+1}\right)=2i+1$, for each $i\in [1,n-1]$;
- 2 .
- $f\left({x}_{n}\right)+f\left({x}_{1}\right)=n+1$;
- 3 .
- $f\left({x}_{i}\right)+f\left({x}_{j}\right)=i+i+2=2i+2$, for $j=(i+2)\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}n$ and $i\in [1,n-2]$.

**Theorem**

**12.**

**Proof.**

- $f({x}_{i})=i$, for every $i\in [1,\frac{n}{2}]$ and $i\in [\frac{n}{2}+3,n]$;
- $f({x}_{\frac{n}{2}+1})=\frac{n}{2}+2$;
- $f({x}_{\frac{n}{2}+2})=\frac{n}{2}+1$.

- $f({x}_{i})+f({x}_{i+1})=2i+1$, for each $i\in [1,\frac{n}{2}-1]$ and $i\in [\frac{n}{2}+2,n-1]$;
- $f({x}_{\frac{n}{2}})+f({x}_{\frac{n}{2}+1})=n+2$;
- $f({x}_{\frac{n}{2}+1})+f({x}_{\frac{n}{2}+2})=n+3$;
- $f({x}_{n})+f({x}_{1})=n+1;$
- $f({x}_{i})+f({x}_{i+2})=2i+2$, for each ${x}_{i}{x}_{i+2}\in E(C{C}_{n}^{2})$, $i\in [1,\frac{n}{2}]$ and $i\in [\frac{n}{2}+3,n-2]$;
- $f({x}_{\frac{n}{2}+1})+f({x}_{\frac{n}{2}+3})=n+5$;
- $f({x}_{\frac{n}{2}-1})+f({x}_{\frac{n}{2}+2})=n$;
- $f({x}_{\frac{n}{2}+1})+f({x}_{(\frac{n}{2}+4)})=n+6$.

**Theorem**

**13.**

- 1 .
- $H\cong Amal({C}_{n};{P}_{n-1};2)$, for any $n\ge 3$;
- 2 .
- $H\cong Amal({C}_{n};{P}_{n-2};2)$, for any $n\ge 3$.

**Proof.**

**Problem**

**3.**

**Lemma**

**6.**

**Proof.**

**Corollary**

**11.**

**Proof.**

## 6. Open Problems

- What are the other forbidden subgraphs of $({K}_{2},{P}_{n})$-sim-(super)magic graph?
- What are the other forbidden subgraphs of $({K}_{2},{C}_{n})$-sim-(super)magic graphs?
- Characterize $({K}_{2},{P}_{n})$-sim-(super)magic graphs for any $n\ge 5$.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

SEMT | Super edge magic total |

EMT | Edge magic total |

## References

- Kotzig, A.; Rosa, A. Magic valuations of finite graphs. Can. Math. Bull.
**1970**, 13, 451–461. [Google Scholar] [CrossRef] - Ringel, G.; Lladó, A. Another tree conjecture. Bull. Inst. Combin. Appl.
**1996**, 18, 83–85. [Google Scholar] - Enomoto, H.; Lladó, A.; Nakamigawa, T.; Ringel, G. Super edge-magic graphs. SUT J. Math.
**1998**, 34, 105–109. [Google Scholar] - Figueroa-Centeno, R.M.; Ichishima, R.; Muntaner-Batle, F.A. The place of super edge magic labelings among other classes of labelings. Discret. Math.
**2001**, 231, 153–168. [Google Scholar] [CrossRef][Green Version] - Bača, M.; Lin, Y.; Miller, M.; Simanjuntak, R. New constructions of magic and antimagic graph labelings. Util. Math.
**2001**, 60, 229–239. [Google Scholar] - Bača, M.; Miller, M.; Ryan, J.; Semaničová-Feňovčíková, A. Magic and Antimagic Graphs: Attributes, Observations, and Challenges in Graph Labelings; Springer International Publishing: Cham, Switzerland, 2019. [Google Scholar]
- Gallian, J.A. A dynamic survey of graph labeling. Electron. J. Combin.
**2020**, 23, DS6. [Google Scholar] - López, S.C.; Muntaner-Batle, F.A. Graceful, Harmonious, and Magic Type Labelings: Relations and Techniques; Springer: Cham, Switzerland, 2017. [Google Scholar]
- Marr, A.M.; Wallis, W.D. Magic Graphs; Birkäuser Basel: Berlin, Germany, 2013. [Google Scholar]
- Macdougall, J.A.; Wallis, W.D. Strong edge-magic graphs of maximum size. Discret. Math.
**2008**, 308, 2756–2763. [Google Scholar] [CrossRef][Green Version] - Sugeng, K.A.; Xie, W. Construction of super edge magic total graphs. In Proceedings of the Sixteenth Australasian Workshop on Combinatorial Algorithms 2005, Ballarat, Australia, 18–21 September 2005; pp. 303–310. [Google Scholar]
- Gutiérrez, A.; Lladó, A. Magic coverings. J. Combin. Math. Combin. Comput.
**2005**, 55, 43–56. [Google Scholar] - Ashari, Y.F.; Salman, A.N.M.; Simanjuntak, R.; Semaničová-Feňovčíková, A.; Bača, M. On (F, H)-simultaneously-magic labeling of graphs. 2021; Submitted. [Google Scholar]
- Maryati, T.K.; Baskoro, E.T.; Salman, A.N.M.; Irawati. On the path-(super)magicness of a cycle with some pendants. Util. Math.
**2015**, 96, 319–330. [Google Scholar] - Maryati, T.K.; Baskoro, E.T.; Salman, A.N.M. P
_{h}-(super)magic labelings of some trees. J. Combin. Math. Combin. Comput.**2008**, 65, 197–204. [Google Scholar] - Lladó, A.; Moragas, J. Cycle-magic graphs. Discret. Math.
**2007**, 307, 2925–2933. [Google Scholar] [CrossRef][Green Version] - Roswitha, M.; Baskoro, E.T.; Maryati, T.K.; Kurdhi, N.A.; Susanti, I. Further results on cycle-supermagic labeling. AKCE Int. J. Graphs Comb.
**2013**, 10, 211–220. [Google Scholar] - Ngurah, A.A.G.; Salman, A.N.M.; Sudarsana, I.W. On supermagic coverings of fans and ladders. SUT J. Math.
**2010**, 46, 67–78. [Google Scholar] - Maryati, T.K.; Salman, A.N.M.; Baskoro, E.T. Supermagic coverings of the disjoint union of graphs and amalgamations. Discret. Math.
**2013**, 313, 397–405. [Google Scholar] [CrossRef] - Philips, N.C.K.; Rees, R.S.; Wallis, W.D. Edge-magic total labelings of wheels. Bull. Inst. Combin. Appl.
**2001**, 31, 21–30. [Google Scholar] - Figueroa-Centeno, R.M.; Ichishima, R.; Muntaner-Batle, F.A. On super edge-magic graphs. Ars Combin.
**2002**, 64, 81–95. [Google Scholar]

**Figure 10.**(

**a**) A $({K}_{2},{P}_{6})$-sim-supermagic labeling of the broom ${B}_{11,6}$ and (

**b**) A $({K}_{2},{P}_{7})$-sim-supermagic labeling of the double broom $D{B}_{14,3,6}$.

**Figure 11.**(

**a**) A ${C}_{4}$-supermagic labeling of ${K}_{4}$ and (

**b**) A ${C}_{4}$-magic labeling of ${K}_{4}$.

**Figure 13.**(

**a**) A $({K}_{2},{C}_{7})$-sim-supermagic labeling $C{C}_{7}^{1}$ and (

**b**) A $({K}_{2},{C}_{8})$-sim-supermagic labeling of $C{C}_{8}^{2}$.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ashari, Y.F.; Salman, A.N.M.; Simanjuntak, R. On Forbidden Subgraphs of (*K*_{2}, *H*)-Sim-(Super)Magic Graphs. *Symmetry* **2021**, *13*, 1346.
https://doi.org/10.3390/sym13081346

**AMA Style**

Ashari YF, Salman ANM, Simanjuntak R. On Forbidden Subgraphs of (*K*_{2}, *H*)-Sim-(Super)Magic Graphs. *Symmetry*. 2021; 13(8):1346.
https://doi.org/10.3390/sym13081346

**Chicago/Turabian Style**

Ashari, Yeva Fadhilah, A.N.M. Salman, and Rinovia Simanjuntak. 2021. "On Forbidden Subgraphs of (*K*_{2}, *H*)-Sim-(Super)Magic Graphs" *Symmetry* 13, no. 8: 1346.
https://doi.org/10.3390/sym13081346