Another H-super magic decompositions of the lexicographic product of graphs

Abstract

<p>Let <span class="math"><em>H</em></span> and <span class="math"><em>G</em></span> be two simple graphs. The concept of an <span class="math"><em>H</em></span>-magic decomposition of <span class="math"><em>G</em></span> arises from the combination between graph decomposition and graph labeling. A decomposition of a graph <span class="math"><em>G</em></span> into isomorphic copies of a graph <span class="math"><em>H</em></span> is <span class="math"><em>H</em></span>-magic if there is a bijection <span class="math"><em>f</em> : <em>V</em>(<em>G</em>) ∪ <em>E</em>(<em>G</em>) → {1, 2, ..., ∣<em>V</em>(<em>G</em>) ∪ <em>E</em>(<em>G</em>)∣}</span> such that the sum of labels of edges and vertices of each copy of <span class="math"><em>H</em></span> in the decomposition is constant. A lexicographic product of two graphs <span class="math"><em>G</em><sub>1</sub></span> and <span class="math"><em>G</em><sub>2</sub>, </span> denoted by <span class="math"><em>G</em><sub>1</sub>[<em>G</em><sub>2</sub>], </span> is a graph which arises from <span class="math"><em>G</em><sub>1</sub></span> by replacing each vertex of <span class="math"><em>G</em><sub>1</sub></span> by a copy of the <span class="math"><em>G</em><sub>2</sub></span> and each edge of <span class="math"><em>G</em><sub>1</sub></span> by all edges of the complete bipartite graph <span class="math"><em>K</em><sub><em>n</em>, <em>n</em></sub></span> where <span class="math"><em>n</em></span> is the order of <span class="math"><em>G</em><sub>2</sub>.</span> In this paper we provide a sufficient condition for <span class="math">$\overline{C_{n}}[\overline{K_{m}}]$</span> in order to have a <span class="math">$P_{t}[\overline{K_{m}}]$</span>-magic decompositions, where <span class="math"><em>n</em> > 3, <em>m</em> > 1, </span> and <span class="math"><em>t</em> = 3, 4, <em>n</em> − 2</span>.</p>

A C3 Magic Decomposition on Friendship Graph with Odd Order

“…Beberapa penelitian sebelumnya telah banyak mengangkat topik penelitian mengenai dekomposisi graf. Contoh penelitian sebelumnya, yaitu pelabelan total titik ajaib pada Graf Peterson yang diperumum (Simangunsong, 2015), dekomposisi H super (anti) ajaib dari graf Antiprisma (Hendy, 2016), dekomposisi super ajaib berbentuk lintasan dari amalgamasi graf siklus oleh Pancahayani (2017) dan graf hasil kali leksikografik (Hendy et al, 2018). Selain itu, telah dikaji pula dekomposisi ( , ) ntiajaib berbentuk pada Graf Generalized Peterson ( , 3) (Masyitoh, 2019) dan.…”

Magic and Antimagic Decomposition of Amalgamation of Cycles

“…Subbiah et al in [4] provide a characterization of 2-factor-edge-super magic decomposable graphs. Moreover, Hendy et al in [5] had investigated a sufficient condition of 𝐻-super magic decompositions of the lexicographic product of cycles and complete graphs. On the other hand, Inayah et al in [6] studied magic and antimagic decomposition for complete graphs, and Hendy et al in [7] explained how to decompose antimagically toroidal grids and triangulations.…”