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On some classes of cycles-related $Γ$-harmonious graphs

Gyaneshwar Agrahari, Dalibor Froncek

Abstract

A graph $G(V,E)$ is $Γ$-harmonious when there is an injection $f$ from $V$ to an Abelian group $Γ$ such that the induced edge labels defined as $w(xy)=f(x)+f(y)$ form a bijection from $E$ to $Γ$. We study $Γ$-harmonious labelings of several cycles-related classes of graphs, including Dutch windmills, generalized prisms, generalized closed and open webs, and superwheels.

On some classes of cycles-related $Γ$-harmonious graphs

Abstract

A graph is -harmonious when there is an injection from to an Abelian group such that the induced edge labels defined as form a bijection from to . We study -harmonious labelings of several cycles-related classes of graphs, including Dutch windmills, generalized prisms, generalized closed and open webs, and superwheels.
Paper Structure (9 sections, 31 theorems, 22 equations, 12 figures)

This paper contains 9 sections, 31 theorems, 22 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Harmonious groups
  4. Known results
  5. Superwheels
  6. Windmill graphs
  7. Generalized prisms
  8. Generalized web graphs
  9. Conclusion

Key Result

Theorem 3.1

Let $\Gamma$ be an Abelian group of order $n=p^{s_1}_1 p^{s_2}_2\dots p^{s_k}_k$, where $k\geq1$, $p_1,p_2,\dots,p_k$ are primes, not necessarily distinct, and $s_1,s_2,\dots,s_k$ positive integers. Then $\Gamma$ is isomorphic to $Z_{p^{s_1}_1}\oplus Z_{p^{s_2}_2}\oplus\dots\oplus Z_{p^{s_k}_k}$ and

Figures (12)

  • Figure 1: Labeling of $SW_{3,5}$ with $Z_{10}\oplus Z_2$
  • Figure 2: Labeling of $SW_{3,5}$ with $Z_4\oplus Z_2\oplus Z_4$
  • Figure 3: (a) $K$-harmonious labeling of each $C^j$ in $D_5^3$ where $K=\Gamma/H$, $\Gamma={Z}_5\oplus {Z}_3$ and $H=\{0\}\oplus {Z}_3$ (b) $\Gamma$-harmonious labeling of $D_5^3$
  • Figure 4: Labeling of $C_5$ with $Z_5$
  • Figure 5: Labeling of $C_{5}$ with $Z_5\oplus \{0\}$
  • ...and 7 more figures

Theorems & Definitions (61)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 3.1: The Fundamental Theorem of Finite Abelian Groups
  • Theorem 3.2
  • ...and 51 more