#
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 |

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**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 $C{C}_{8}^{2}$.

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**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