$H$-$V$-Super-Strong-$(a,d)$-antimagic decomposition of complete bipartite graphs

Kumar, Solomon

doi:10.61091/ars157-07

Abstract

References

Ars Combinatoria

Volume 157
Pages: 73-80

Research article

$H$ - $V$ -Super-Strong- $(a, d)$ -antimagic decomposition of complete bipartite graphs

^¹

¹Department of Mathematics, The American College, Madurai – 625 002, Tamilnadu, India.

Received: 26/07/2023
Accepted: 07/12/2023
Published: 13/12/2023

Copyright Link
License

Abstract

An $H$ -(a,d)-antimagic labeling in a $H$ -decomposable graph $G$ is a bijection $f : V (G) \cup E (G) \to {1, 2, \dots, p + q}$ such that $\sum f (H_{1}), \sum f (H_{2}), \dots, \sum f (H_{h})$ forms an arithmetic progression with difference $d$ and first element $a$ . $f$ is said to be $H$ - $V$ -super- $(a, d)$ -antimagic if $f (V (G)) = {1, 2, \dots, p}$ . Suppose that $V (G) = U (G) \cup W (G)$ with $| U (G) | = m$ and $| W (G) | = n$ . Then $f$ is said to be $H$ - $V$ -super-strong- $(a, d)$ -antimagic labeling if $f (U (G)) = {1, 2, \dots, m}$ and $f (W (G)) = {m + 1, m + 2, \dots, (m + n = p)}$ . A graph that admits a $H$ - $V$ -super-strong- $(a, d)$ -antimagic labeling is called a $H$ - $V$ -super-strong- $(a, d)$ -antimagic decomposable graph. In this paper, we prove that complete bipartite graphs $K_{m, n}$ are $H$ - $V$ -super-strong- $(a, d)$ -antimagic decomposable with both $m$ and $n$ are even.

Keywords:

H

-decomposable graph,

H

V

-super magic labeling, Complete bipartite graph

1. Introduction

In this paper we consider only finite and simple undirected bipartite graphs. The vertex and edge sets of a graph G are denoted by $V (G)$ and $E (G)$ respectively and we let $| V (G) | = p$ and $| E (G) | = q$ . For graph theoretic notations, we follow [1, 2]. A labeling of a graph G is a mapping that carries a set of graph elements, usually vertices and/or edges into a set of numbers, usually integers. Many kinds of labeling have been studied and an excellent survey of graph labeling can be found in[3].

Although magic labeling of graphs was introduced by Sedlacek [4], the concept of vertex magic total labeling (VMTL) first appeared in 2002 in[5]. In 2004, MacDougall et al. [6] introduced the notion of super vertex magic total labeling (SVMTL). In 1998, Enomoto et al. [7] introduced the concept of super edge-magic graphs. In 2005, Sugeng and Xie[8] constructed some super edge-magic total graphs. The usage of the word “super” was introduced in[7]. The notion of a $V$ -super vertex magic labeling was introduced by MacDougall et al.[6] as in the name of super vertex-magic total labeling and it was renamed as $V$ -super vertex magic labeling by Marr and Wallis in [9] after referencing the article[10]. Most recently, Tao-ming Wang and Guang-Hui Zhang [11], generalized some results found in [10].

Hartsfield and Ringel [12] introduced the concept of an antimagic graph. In their terminology, an antimagic labeling is an edge-labeling of the graph with the integers $1, 2, \dots, q$ so that the weight at each vertex is different from the weight at any other vertex. Bodendiek and Walther [13] defined the concept of an $(a, d)$ -antimagic labeling as an edge-labeling in which the vertex weights forms an arithmetic progression starting from $a$ and having common difference $d$ . B $\overset{ˇ}{a}$ ca et al.[14] introduced the notions of vertex-antimagic total labeling and $(a, d)$ -vertex-antimagic total labeling. Simanjuntak et al [15] introduced the concept of $(a, d)$ -antimagic graph. Sudarasana et al [16] studied the concept of super edge-antimagic total lableing of disconnected graphs.

A bijection $f$ from $V (G) \cup E (G)$ to the integers $1, 2, \dots, p + q$ is called a vertex-antimagic total labeling of $G$ if the weights of vertices ${w_{f} (x) = f (x) + \sum_{x y \in E (G)} f (x y)$ , $x \in V (G)$ }, are pairwise distinct. $f$ is called an $(a, d)$ -vertex-antimagic total labeling of $G$ if the set of vertex weights ${w_{f} (x) | x \in V (G)} = {a, a + d, \dots, a + (p - 1) d}$ for some integers $a$ and $d$ . $f$ is said to be super- $(a, d)$ -vertex-antimagic labeling if $f (V (G)) = {1, 2, \dots, p}$ . A graph $G$ is called super- $(a, d)$ -vertex-antimagic if it admits a super- $(a, d)$ -vertex-antimagic labeling. A bijection $f$ from $V (G) \cup E (G)$ to the integers $1, 2, \dots, p + q$ is called an $(a, d)$ -edge-antimagic total labeling of $G$ if the edge weights ${w (u v) = f (u) + f (v) + f (u v)$ , $u v \in E (G)$ }, forms an arithmetic sequence with the first term $a$ and common difference $d$ . $f$ is said to be super- $(a, d)$ -edge-antimagic labeling if $f (V (G)) = {1, 2, \dots, p}$ . A graph $G$ is called super- $(a, d)$ -edge-antimagic if it admits a super- $(a, d)$ -edge-antimagic labeling.

A covering of $G$ is a family of subgraphs $H_{1}, H_{2}, \dots, H_{h}$ such that each edge of $E (G)$ belongs to at least one of the subgraphs $H_{i}$ , $1 \leq i \leq h$ . Then it is said that $G$ admits an $(H_{1}, H_{2}, \dots, H_{h})$ covering. If every $H_{i}$ is isomorphic to a given graph $H$ , then $G$ admits an $H$ -covering. A family of subgraphs $H_{1}, H_{2}, \dots, H_{h}$ of $G$ is a $H$ -decomposition of $G$ if all the subgraphs are isomorphic to a graph $H$ , $E (H_{i}) \cap E (H_{j}) = \emptyset$ for $i \neq j$ and $\cup_{i = 1}^{h} E (H_{i}) = E (G)$ . In this case, we write $G = H_{1} \oplus H_{2} \oplus \dots \oplus H_{h}$ and $G$ is said to be $H$ -decomposable.

The notion of $H$ -super magic labeling was first introduced and studied by Guti $\overset{´}{e}$ rrez and Llad $\overset{´}{o}$ [17] in 2005. They proved that some classes of connected graphs are $H$ -super magic. Suppose $G$ is $H$ -decomposable. A total labeling $f : V (G) \cup E (G) \to {1, 2, \dots, p + q}$ is called an $H$ -magic labeling of $G$ if there exists a positive integer $k$ (called magic constant) such that for every copy $H$ in the decomposition, $\sum_{v \in V (H)} f (v) + \sum_{e \in E (H)} f (e) = k$ . A graph $G$ that admits such a labeling is called a $H$ -magic decomposable graph. An $H$ -magic labeling $f$ is called a $H$ – $V$ -super magic labeling if $f (V (G)) = {1, 2, \dots, p}$ . A graph that admits a $H$ – $V$ -super magic labeling is called a $H$ – $V$ -super magic decomposable graph. An $H$ -magic labeling $f$ is called a $H$ – $E$ -super magic labeling if $f (E (G)) = {1, 2, \dots, q}$ . A graph that admits a $H$ – $E$ -super magic labeling is called a $H$ – $E$ -super magic decomposable graph. The sum of all vertex and edge labels on $H$ is denoted by $\sum f (H)$ .

In 2001, Muntaner-Batle[18] introduced the concept of super-strong magic labeling of bipartite graph as in the name of special super magic labeling of bipartite graph and it was renamed as super-strong magic labeling by Marr and Wallis[9]. Marimuthu and Stalin Kumar [19] introduced the concept of $H$ – $V$ -super-strong magic decomposition and $H$ – $E$ -super-strong magic decomposition of complete bipartite graphs. Suppose $G$ is a bipartite graph with vertex-sets $V_{1}$ and $V_{2}$ of sizes $m$ and $n$ respectively. An edge-magic total labeling of $G$ is super-strong if the elements of $V_{1}$ receive labels ${1, 2, \dots, m}$ and the elements of $V_{2}$ receive labels ${m + 1, m + 2, \dots, m + n}$ . Suppose $G$ is $H$ -decomposable and if $V (G) = U (G) \cup W (G)$ with $| U (G) | = m$ and $| W (G) | = n$ . An $H$ – $V$ -super magic labeling $f$ is called a $H$ – $V$ -super-strong magic if $f (U (G)) = {1, 2, \dots, m}$ and $f (W (G)) = {m + 1, m + 2, \dots, (m + n = p)}$ . A graph that admits a $H$ – $V$ -super-strong magic labeling is called a $H$ – $V$ -super-strong magic decomposable graph. An $H$ – $E$ -super magic labeling $f$ is called a $H$ – $E$ -super-strong magic labeling if if $f (U (G)) = {q + 1, q + 2, \dots, q + m}$ and $f (W (G)) = {q + m + 1, q + m + 2, \dots, (q + m + n = q p)}$ . A graph that admits a $H$ – $E$ -super-strong magic labeling is called a $H$ – $E$ -super-strong magic decomposable graph.

Suppose $G$ is $H$ -decomposable. A total labeling $f : V (G) \cup E (G) \to {1, 2, \dots, p + q}$ is called an $H$ -antimagic labeling of $G$ if $\sum f (H_{1}), \sum f (H_{2}), \dots, \sum f (H_{h})$ are pairwaise distinct. $f$ is said to be $H$ – $(a, d)$ -antimagic if these numbers forms an arithmetic progression with difference $d$ and first element $a$ . A $H$ – $(a, d)$ -antimagic labeling $f$ is called $H$ – $V$ -super- $(a, d)$ -antimagic labeling if $f (V (G)) = {1, 2, \dots, p}$ . Suppose that $V (G) = U (G) \cup W (G)$ with $| U (G) | = m$ and $| W (G) | = n$ . Then $f$ is said to be $H$ – $V$ -super-strong- $(a, d)$ -antimagic labeling if $f (U (G)) = {1, 2, \dots, m}$ and $f (W (G)) = {m + 1, m + 2, \dots, (m + n = p)}$ . A graph that admits a $H$ – $V$ -super-strong- $(a, d)$ -antimagic labeling is called a $H$ – $V$ -super-strong- $(a, d)$ -antimagic decomposable graph. A $H$ – $(a, d)$ -antimagic labeling $f$ is called $H$ – $E$ -super- $(a, d)$ -antimagic labeling if $f (E (G)) = {1, 2, \dots, q}$ . $f$ is said to be $H$ – $E$ -super-strong- $(a, d)$ -antimagic labeling if $f (U (G)) = {q + 1, q + 2, \dots, q + m}$ and $f (W (G)) = {q + m + 1, q + m + 2, \dots, (q + m + n = q p)}$ . A graph that admits a $H$ – $E$ -super-strong- $(a, d)$ -antimagic labeling is called a $H$ – $E$ -super-strong- $(a, d)$ -antimagic decomposable graph.

In 2012, Inayah et al. [20] studied magic and anti-magic $H$ -decompositions and Zhihe Liang [21] studied cycle-super magic decompositions of complete multipartite graphs. In many of the results about $H$ -magic graphs, the host graph $G$ is required to be $H$ -decomposable. Yoshimi Ecawa et al [22] studied the decomposition of complete bipartite graphs into edge-disjoint subgraphs with star components. The notion of star-subgraph was introduced by Akiyama and Kano in [23]. A subgraph $F$ of a graph $G$ is called a star-subgraph if each component of $F$ is a star. Here by a star, we mean a complete bipartite graph of the form $K_{1, m}$ with $m \geq 1$ . A subgraph $F$ of a graph $G$ is called a $n$ -star-subgraph if $F ≅ K_{1, n}$ with $2 \leq n < p$ . Marimuthu and Stalin Kumar [24, 25] studied about the $H$ – $V$ -super magic decomposition and $H$ – $E$ -super magic decomposition of complete bipartite graphs.

2. Main Results

In this section, we consider the graphs $G ≅ K_{m, n}$ and $H ≅ K_{1, n}$ , where $n \geq 1$ and both $m$ and $n$ are even. Clearly $p = m + n$ and $q = m n$ .

Theorem 1. Suppose ${H_{1}, H_{2}, \dots, H_{m}}$ is a $n$ -star-decomposition of $G$ with both $m$ and $n$ are even. Then $G$ is $H$ – $V$ -super-strong- $(a, d)$ -antimagic decomposable with $a = 1 + \frac{n^{2} (m + 3) + 2 n (2 m + 1)}{2}$ and $d = 1$ .

Proof. Let $U = {u_{1}, u_{2}, \dots, u_{m}}$ and $V = {v_{1}, v_{2}, \dots, v_{n}}$ be two stable sets of $G$ . Let ${H_{1}, H_{2}, \dots, H_{m}}$ be a $n$ -star decomposition of $G$ with both $m$ and $n$ are even, where each $H_{i}$ is isomorphic to $H$ , such that $V (H_{i}) = {u_{i}, v_{1}, v_{2}, \dots, v_{n}}$ and $E (H_{i}) = {u_{i} v_{1}, u_{i} v_{2}, \dots, u_{i} v_{n}}$ , for all $1 \leq i \leq m$ . Define a total labeling $f : V (G) \cup E (G) \to {1, 2, \dots, p + q}$ by $f (u_{i}) = i$ and $f (v_{j}) = m + j$ , for all $1 \leq i \leq m$ and $1 \leq j \leq n$ .

Case 1: $m \neq n$ .
Now the edges of $G$ can be labeled as shown in Table $1$ .

The edge label of a $n$ -star-decomposition of $G$ if $m \neq n$ ..
$f$	$v_{1}$	$v_{2}$	$\dots$	$v_{n - 1}$	$v_{n}$
$u_{1}$	$(m + n)$	$(2 m + n)$	$\dots$	$(m + n)$	$(m + n)$
	$+ m$	$+ 1$		$+ ((n - 1) m)$	$+ ((n - 1) m + 1)$
$u_{2}$	$(m + n) +$	$(2 m + n)$	$\dots$	$(m + n)$	$(m + n)$
	$(m - 1)$	$+ 2$		$+ ((n - 1) m - 1)$	$+ ((n - 1) m + 2)$
$u_{3}$	$(m + n) +$	$(2 m + n)$	$\dots$	$(m + n)$	$(m + n)$
	$(m - 2)$	$+ 3$		$((n - 1) m - 2)$	$+ ((n - 1) m + 3)$
$⋮$	$\dots$	$\dots$	$\dots$	$\dots$	$\dots$
$u_{k}$	$(m + n) +$	$(2 m + n)$	$\dots$	$(m + n) + ((n - 2) m)$	$(m + n) + (n - 1) m$
	$(m - (k - 1))$	$+ k$		$+ (m - (k - 1))$	$+ k$
$⋮$	$\dots$	$\dots$	$\dots$	$\dots$	$\dots$
$u_{m - 1}$	$(m + n) +$	$(2 m + n)$	$\dots$	$(m + n)$	$(m + n)$
	$2$	$+ (m - 1)$		$+ ((n - 2) m + 2)$	$+ (m n - 1)$
$u_{m}$	$(m + n) +$	$(2 m + n)$	$\dots$	$(m + n)$	$(m + n)$
	$1$	$+ m$		$+ ((n - 2) m + 1)$	$+ m n$

We prove the result for $n = k$ and the result follows for all $1 \leq k \leq m$ .
From Table $1$ and from definition of $f$ , we get $\begin{aligned} \sum f (H_{k}) & = & f (u_{k}) + \sum_{i = 1}^{n} f (v_{i}) + \sum_{i = 1}^{n} f (u_{k} v_{i}) = k + \sum_{i = 1}^{n} (m + i) + \sum_{i = 1}^{n} f (u_{k} v_{i}) . \end{aligned}$ Now, $\begin{aligned} \sum_{i = 1}^{n} f (v_{i}) & = & (m + 1) + (m + 2) + \dots + (m + n) \\ = & m n + (1 + 2 + \dots + n) = m n + \frac{n (n + 1)}{2} . \end{aligned}$ Also $\begin{aligned} \sum_{i = 1}^{n} f (u_{k} v_{i}) & = & ((m + n) + (m - (k - 1))) + ((m + n) + (m + k)) + \dots \\ + ((m + n) + (n - 2) m + (m - (k - 1))) + ((m + n) + (n - 1) m + k) \\ = & ((2 m + n) - (k - 1)) + ((2 m + n) + k) + ((4 m + n) - (k - 1)) + \\ ((4 m + n) + k) + \dots + (((n) m + n) - (k - 1)) + (((n) m + n) + k) \\ = & 2 ((2 m + n) + (4 m + n) + \dots + (n m + n)) + \frac{n}{2} (1) \\ = & 2 ((2 m + 2 n + \dots + n m) + \frac{n (n)}{2}) + \frac{n}{2} \\ = & 4 m (1 + 2 + \dots + \frac{n}{2}) + \frac{2 n^{2} + n}{2} = 4 m (\frac{n (n + 2)}{8}) + \frac{2 n^{2} + n}{2} \\ = & \frac{m n^{2} + 2 m n + 2 n^{2} + n}{2} = \frac{n^{2} (m + 2) + n (2 m + 1)}{2} . \end{aligned}$
Hence $\begin{aligned} \sum_{i = 1}^{n} f (u_{k} v_{i}) & = & \frac{n^{2} (m + 2) + n (2 m + 1)}{2} . \end{aligned}$ and is constant for all $1 \leq k \leq m$ .
Using the above values, we get $\begin{aligned} \sum f (H_{k}) & = & k + m n + \frac{n (n + 1)}{2} + \frac{n^{2} (m + 2) + n (2 m + 1)}{2} \\ = & k + \frac{2 m n + n^{2} + n + n^{2} (m + 2) + n (2 m + 1)}{2} \\ = & k + \frac{n^{2} (m + 3) + 2 n (2 m + 1)}{2} . \end{aligned}$ for all $1 \leq k \leq m$ . So, ${\sum f (H_{1}), \sum f (H_{2}), \dots, \sum f (H_{m}) = a, a + d, \dots, a + (m - 1) d}$ forms an arithmetic progression with $a = (1 + \frac{n^{2} (m + 3) + 2 n (2 m + 1)}{2})$ and common difference $d = 1$ . Thus in this case, the graph $G$ is a $H$ – $V$ -super-strong- $(a, d)$ -antimagic decomposable.

Case 2: $m = n$ .
Now the edges of $G$ can be labeled as shown in Table $2$ .

The edge label of a $n$ -star-decomposition of $G$ if $m = n$ .
$f$	$v_{1}$	$v_{2}$	$\dots$	$v_{n - 1}$	$v_{n}$
$u_{1}$	$3 n$	$3 n + 1$	$\dots$	$(n + 1) n$	$(n + 1) n + 1$
$u_{2}$	$3 n - 1$	$3 n + 2$	$\dots$	$(n + 1) n - 1$	$(n + 1) n + 2$
$u_{3}$	$3 n - 2$	$3 n + 3$	$\dots$	$(n + 1) n - 2$	$(n + 1) n + 3$
$⋮$	$\dots$	$\dots$	$\dots$	$\dots$	$\dots$
$u_{k}$	$3 n - (k - 1)$	$3 n + k$	$\dots$	$(n + 1) n - (k - 1)$	$(n + 1) n + k$
$⋮$	$\dots$	$\dots$	$\dots$	$\dots$	$\dots$
$u_{n - 1}$	$2 n + 2$	$4 n - 1$	$\dots$	$n (n) + 2$	$(n + 2) n - 1$
$u_{n}$	$2 n + 1$	$4 n$	$\dots$	$n (n) + 1$	$(n + 2) n$

We prove the result for $n = k$ and the result follows for all $1 \leq k \leq n$ .
From Table $2$ and from definition of $f$ , we get $\begin{aligned} \sum f (H_{k}) & = & f (u_{k}) + \sum_{i = 1}^{n} f (v_{i}) + \sum_{i = 1}^{n} f (u_{k} v_{i}) = k + \sum_{i = 1}^{n} (n + i) + \sum_{i = 1}^{n} f (u_{k} v_{i}) . \end{aligned}$ Now, $\begin{aligned} \sum_{i = 1}^{n} f (v_{i}) & = & (n + 1) + (n + 2) + \dots + (n + n) = (n) n + (1 + 2 + \dots + n) \\ = & (n) n + \frac{n (n + 1)}{2} . \end{aligned}$ Also $\begin{aligned} \sum_{i = 1}^{n} f (u_{k} v_{i}) & = & (3 n - (k - 1)) + (3 n + k) + (5 n - (k - 1)) + (5 n + k) + \dots \\ + ((n + 1) n - (k - 1)) + ((n + 1) n + k) \\ = & (3 n + 1) + 3 n + (5 n + 1) + 5 n + \dots + ((n + 1) n + 1) + (n + 1) n \\ = & 2 (3 n + 5 n + \dots + (n + 1) n) + \frac{n}{2} (1) \\ = & 2 n (3 + 5 + \dots + (n + 1)) + \frac{n}{2} \\ = & 2 n ((1 + 2 + 3 + \dots + (n + 1)) - (2 + 4 + 6 + \dots + n) - 1) + \frac{n}{2} \\ = & 2 n (\frac{(n + 1) (n + 2)}{2} - 2 \frac{(\frac{n}{2}) (\frac{n + 1}{2})}{2} - 1) + \frac{n}{2} \\ = & 2 (\frac{n^{2} + 3 n + 2}{2} - \frac{(n^{2} + 2 n)}{4} - 1) + \frac{n}{2} \\ = & 2 n (\frac{2 n^{2} + 6 n + 4 - n^{2} - 2 n - 4}{4} + \frac{n}{2} = \frac{n (n^{2} + 4 n + n)}{2} \\ = & \frac{n^{3} + 2 n^{2} + 2 n^{2} + n}{2} = \frac{n^{2} (n + 2) + (n (2 n + 1)}{2} . \end{aligned}$ Hence $\begin{aligned} \sum_{i = 1}^{n} f (u_{k} v_{i}) & = & \frac{n^{2} (n + 2) + n (2 n + 1)}{2} . \end{aligned}$ and is constant for all $1 \leq k \leq n$ .
Using the above values, we get $\begin{aligned} \sum f (H_{k}) & = & k + (n) n + \frac{n (n + 1)}{2} + \frac{n^{2} (n + 2) + n (2 n + 1)}{2} \\ = & k + \frac{2 (n) n + n^{2} + n + n^{2} (n + 2) + n (2 n + 1)}{2} \\ = & k + \frac{n^{2} (n + 3) + 2 n (2 n + 1)}{2} . \end{aligned}$ for all $1 \leq k \leq n$ . So, ${\sum f (H_{1}), \sum f (H_{2}), \dots, \sum f (H_{n}) = a, a + d, \dots, a + (n - 1) d}$ forms an arithmetic progression with $a = (1 + \frac{n^{2} (n + 3) + 2 n (2 n + 1)}{2})$ and common difference $d = 1$ . Thus in this case also, the graph $G$ is a $H$ – $V$ -super-strong- $(a, d)$ -antimagic decomposable.

Theorem 2. If a non-trivial $H$ -decomposable graph $G ≅ K_{m, n}$ is $H$ – $V$ -super-strong- $(a, d)$ -antimagic decomposable graph with both $m$ and $n$ are even and if the sum of edge labels of a decomposition $H_{j}$ is denoted by $\sum f (E (H_{j}))$ then $\sum f (E (H_{j}))$ is constant for all $1 \leq j \leq m$ and it is given by $\sum f (E (H_{j})) = \frac{n^{2} (m + 2) + n (2 m + 1)}{2}$ .

Proof. Suppose $G$ is $H$ -decomposable and possesses a $H$ – $V$ -super-strong- $(a, d)$ -antimagic labeling $f$ , then by Theorem 1, for each $H_{j}$ in the $H$ -decomposition of $G$ , we get $\begin{aligned} \sum f (E (H_{j})) & = & \sum_{i = 1}^{n} f (u_{j} v_{i}) = \frac{n^{2} (m + 2) + n (2 m + 1)}{2} \end{aligned}$ which is true for all $1 \leq j \leq m$ . Thus $\sum f (E (H_{j}))$ is constant for all $1 \leq k \leq m$ and it is given by $\sum f (E (H_{j})) = \frac{n^{2} (m + 2) + n (2 m + 1)}{2}$ .

Theorem 3. If a non-trivial $H$ -decomposable graph $G ≅ K_{m, n}$ is $H$ – $V$ -super-strong- $(a, d)$ -antimagic decomposable graph with both $m$ and $n$ are even and if the sum of vertex labels of a decomposition $H_{j}$ is denoted by $\sum f (V (H_{j}))$ then
${\sum f (V (H_{1})), \sum f (V (H_{2})), \dots, \sum f (V (H_{m}))} = {a, a + d, \dots, a + (m - 1) d}$ with $a = (m n + 1) + \frac{n (n + 1)}{2}$ and $d = 1$ .

Proof. Suppose $G$ is $H$ -decomposable and possesses a $H$ – $V$ -super-strong- $(a, d)$ -antimagic labeling $f$ , then by Theorem 1, for each $H_{j}$ in the $H$ -decomposition of $G$ , we get $\begin{aligned} \sum f (V (H_{j})) & = & f (u_{j}) + \sum_{i = 1}^{n} f (v_{i}) = j + \sum_{i = 1}^{n} (m + i) = j + ((m + 1) + (m + 2) + \dots + (m + n)) \\ = & j + m n + \frac{n (n + 1)}{2} . \end{aligned}$ which is true for all $1 \leq j \leq m$ . Thus ${\sum f (V (H_{1})), \sum f (V (H_{2})), \dots, \sum f (V (H_{m}))} = {a, a + d, \dots, a + (m - 1) d}$ with $a = (m n + 1) + \frac{n (n + 1)}{2}$ and $d = 1$ .

Theorem 4. Let $G ≅ K_{m, n}$ be a $H$ -decomposable graph with both $m$ and $n$ are even and if $V (G) = U (G) \cup W (G)$ with $| U (G) | = m$ and $| W (G) | = n$ . let $g$ be a bijection from $V (G)$ onto ${1, 2, \dots, p}$ with $g (U (G)) = {1, 2, \dots, m}$ and $g (W (G)) = {(m + 1), (m + 2), \dots, (m + n = p)}$ then $g$ can be extended to an $H$ – $V$ -super-strong- $(a, d)$ -antimagic labeling if and only if $\sum f (E (H_{j}))$ is constant for all $1 \leq j \leq m$ and it is given by $\sum f (E (H_{j})) = \frac{n^{2} (m + 2) + n (2 m + 1)}{2}$ .

Proof. Suppose $G ≅ K_{m . n}$ be a $H$ -decomposable graph with both $m$ and $n$ are even and if $V (G) = U (G) \cup W (G)$ with $| U (G) | = m$ and $| W (G) | = n$ . let $g$ be a bijection from $V (G)$ onto ${1, 2, \dots, p}$ with $g (U (G)) = {1, 2, \dots, m}$ and $g (W (G)) = {(m + 1), (m + 2), \dots, (m + n = p)}$ . Assume that $\sum f (E (H_{j}))$ is constant for all $1 \leq j \leq m$ and it is given by $\sum f (E (H_{j})) = \frac{n^{2} (m + 2) + n (2 m + 1)}{2}$ . Define $f : V (G) \cup E (G) \to {1, 2, \dots, p + q}$ as $f (u_{i}) = g (u_{i})$ ; $f (u_{j}) = g (u_{j})$ for all $1 \leq i \leq m$ ; $1 \leq j \leq n$ and the edge labels are in either Table $1$ (if $m \neq n$ ) or Table $2$ (if $m = n$ ) then by Theorem $2.1$ , for each $H_{j}$ in the $H$ -decomposition of $G$ , we get $\begin{aligned} \sum f (V (H_{j})) & = & f (u_{j}) + \sum_{i = 1}^{n} f (v_{i}) = j + \sum_{i = 1}^{n} (m + i) = j + ((m + 1) + (m + 2) + \dots + (m + n)) \\ = & j + m n + \frac{n (n + 1)}{2} . \end{aligned}$ which is true for all $1 \leq j \leq m$ . So, we have ${\sum f (V (H_{1})), \sum f (V (H_{2})), \dots, \sum f (V (H_{m}))} = {a, a + d, \dots, a + (m - 1) d}$ with $a = (m n + 1) + \frac{n (n + 1)}{2}$ and $d = 1$ . Hence, $\begin{aligned} \sum f (H_{j}) & = & \sum f (V (H_{j})) + \sum f (E (H_{j})) = (j + m n + \frac{n (n + 1)}{2}) + (\frac{n^{2} (m + 2) + n (2 m + 1)}{2}) \\ = & j + \frac{2 m n + n^{2} + n + n^{2} (m + 2) + n (2 m + 1)}{2} = j + \frac{n^{2} (m + 3) + 2 n (2 m + 1)}{2} . \end{aligned}$ for every $H_{j}$ in the $H$ -decomposition of $G$ and for all $1 \leq j \leq m$ . Thus we have, $f$ is an $H$ – $V$ -super-strong- $(a, d)$ -antimagic labeling.
Suppose $g$ can be extended to an $H$ – $V$ -super-strong- $(a, d)$ -antimagic labeling $f$ of $G$ with with $a = 1 + \frac{n^{2} (m + 3) + 2 n (2 m + 1)}{2}$ and $d = 1$ . Then by Theorem 2 $\sum f (E (H_{j}))$ is constant for all $1 \leq j \leq m$ and it is given by $\sum f (E (H_{j})) = \frac{n^{2} (m + 2) + n (2 m + 1)}{2}$ .

3. Conclusion

In this paper, we studied the $H$ – $V$ -super-strong- $(a, d)$ -antimagic decomposition of $K_{m, n}$ with $n \geq 1$ and both $m$ and $n$ are even.

Conflict of Interest

The author declares no conflict of interests.

References:

Chartrand, G. and Lesniak, L., 1996. Graphs and Digraphs. Chapman and Hall, Boca Raton, London, Newyork, Washington, D.C.[Google Scholor]
Chartrand, G. and Zhang, P., 2009. Chromatic graph theory. Chapman and Hall, CRC, Boca Raton.[Google Scholor]
Gallian, J.A., 2018. A dynamic survey of graph labeling. Electronic Journal of combinatorics, 1(Dynamic Surveys), p.DS6.[Google Scholor]
Sedláček, J., 1963, June. Problem 27. Theory of graphs and its applications. In Proceedings of the Symposium held in Smolenice. Praha (pp. 163-164).[Google Scholor]
MacDougall, J.A., Miller, M. and Wallis, W.D., 2002. Vertex-magic total labelings of graphs. Utilitas Mathematics, 61, pp.3-21.[Google Scholor]
MacDougall, J.A., Miller, M. and Sugeng, K.A., 2004. Super vertex-magic total labelings of graphs. In Proceedings of the 15th Australasian Workshop on Combinatorial Algorithms (pp. 222-229).[Google Scholor]
Enomoto, H., Llado, A.S., Nakamigawa, T. and Ringel, G., 1998. Super edge-magic graphs. SUT Journal of Mathematics, 34(2), pp.105-109.[Google Scholor]
Sugeng, K.A. and Xie, W., 2005. Construction of super edge magic total graphs. Proceedings. 16th AWOCA, 303-310.[Google Scholor]
Marr, A.M. and Wallis, W.D., 2013. Magic graphs(2nd edition). Birkhauser, Boston, Basel, Berlin./li>
Marimuthu, G. and Balakrishnan, M., 2012. E-super vertex magic labelings of graphs. Discrete Applied Mathematics, 160(12), pp.1766-1774.[Google Scholor]
Wang, T.M. and Zhang, G.H., 2014. Note on E-super vertex magic graphs. Discrete Applied Mathematics, 178, pp.160-162.[Google Scholor]
Ringel, G. and Hartsfield, N., 1990. Pearls in graph theory. Academic Press, Boston-SanDiego-Newyork-London.[Google Scholor]
Bodendiek, R. and Walther, G., 1993. Arithmetisch antimagische graphen. Graphentheorie III. Ink. Wagner and R. Bodendiek, Bl-wiss. Verl., Manheim[Google Scholor]
Baca, M., MacDougall, J., Bertault, F., Miller, M., Simanjuntak, R. and Slamin, , 2003. Vertex-antimagic total labelings of graphs. Discussiones mathematicae graph theory, 23(1), pp.67-83.[Google Scholor]
Simanjuntak, R., Bertault, F. and Miller, M., 2000. Two new (a, d)-antimagic graph labelings. In Proceedings of Eleventh Australasian Workshop on Combinatorial Algorithms (Vol. 11, pp. 179-189).[Google Scholor]
Sudarsana, I.W., Ismaimuza, D., Baskoro, E.T. and Assiyatun, H., 2005. On super (a, d)-edge antimagic total labeling of disconnected graphs. Journal of Combinatorial Mathematics and Combinatorial Computing, 55, pp.149-158.[Google Scholor]
Gutiérrez, A. and Lladó, A., 2005. Magic coverings. Journal of Combinatorial Mathematics and Combinatorial Computing, 55, 43-56.[Google Scholor]
Muntaner-Batle, F.A., 2001. Special Super Edge Magic-Labelings of Bipartite Graphs. Journal of Combinatorial Mathematics and Combinatorial Computing, 39, pp.107-120.[Google Scholor]
Kumar, S. S., and Marimuthu, G. A H-V-super-strong magic decomposition of complete bipartite graphs, communicated.
Inayah, N., Lladó, A. and Moragas, J., 2012. Magic and antimagic H-decompositions. Discrete Mathematics, 312(7), pp.1367-1371.[Google Scholor]
Liang, Z., 2012. Cycle-supermagic decompositions of complete multipartite graphs. Discrete Mathematics, 312(22), pp.3342-3348.[Google Scholor]
Egawa, Y., Urabe, M., Fukuda, T. and Nagoya, S., 1986. A decomposition of complete bipartite graphs into edge-disjoint subgraphs with star components. Discrete mathematics, 58(1), pp.93-95.[Google Scholor]
Akiyama, J. and Kano, M., 1984. Path factors of a graph, Graphs and Applications, Wiley, Newyork.[Google Scholor]
Marimuthu, G.T. and Kumar, S.S., H-E-super magic decomposition of complete bipartite graphs, communicated.[Google Scholor]
Kumar, S.S. and Marimuthu, G.T., 2015. HV-super magic decomposition of complete bipartite graphs. Communications of the Korean Mathematical Society, 30(3), pp.313-325.[Google Scholor]

[1] Chartrand, G. and Lesniak, L., 1996. Graphs and Digraphs. Chapman and Hall, Boca Raton, London, Newyork, Washington, D.C.[Google Scholor]

[2] Chartrand, G. and Zhang, P., 2009. Chromatic graph theory. Chapman and Hall, CRC, Boca Raton.[Google Scholor]

[13] Gallian, J.A., 2018. A dynamic survey of graph labeling. Electronic Journal of combinatorics, 1(Dynamic Surveys), p.DS6.[Google Scholor]

[4] Sedláček, J., 1963, June. Problem 27. Theory of graphs and its applications. In Proceedings of the Symposium held in Smolenice. Praha (pp. 163-164).[Google Scholor]

[5] MacDougall, J.A., Miller, M. and Wallis, W.D., 2002. Vertex-magic total labelings of graphs. Utilitas Mathematics, 61, pp.3-21.[Google Scholor]

[6] MacDougall, J.A., Miller, M. and Sugeng, K.A., 2004. Super vertex-magic total labelings of graphs. In Proceedings of the 15th Australasian Workshop on Combinatorial Algorithms (pp. 222-229).[Google Scholor]

[7] Enomoto, H., Llado, A.S., Nakamigawa, T. and Ringel, G., 1998. Super edge-magic graphs. SUT Journal of Mathematics, 34(2), pp.105-109.[Google Scholor]

[8] Sugeng, K.A. and Xie, W., 2005. Construction of super edge magic total graphs. Proceedings. 16th AWOCA, 303-310.[Google Scholor]

[9] Marr, A.M. and Wallis, W.D., 2013. Magic graphs(2nd edition). Birkhauser, Boston, Basel, Berlin./li>

[10] Marimuthu, G. and Balakrishnan, M., 2012. E-super vertex magic labelings of graphs. Discrete Applied Mathematics, 160(12), pp.1766-1774.[Google Scholor]

[11] Wang, T.M. and Zhang, G.H., 2014. Note on E-super vertex magic graphs. Discrete Applied Mathematics, 178, pp.160-162.[Google Scholor]

[12] Ringel, G. and Hartsfield, N., 1990. Pearls in graph theory. Academic Press, Boston-SanDiego-Newyork-London.[Google Scholor]

[13] Bodendiek, R. and Walther, G., 1993. Arithmetisch antimagische graphen. Graphentheorie III. Ink. Wagner and R. Bodendiek, Bl-wiss. Verl., Manheim[Google Scholor]

[14] Baca, M., MacDougall, J., Bertault, F., Miller, M., Simanjuntak, R. and Slamin, , 2003. Vertex-antimagic total labelings of graphs. Discussiones mathematicae graph theory, 23(1), pp.67-83.[Google Scholor]

[15] Simanjuntak, R., Bertault, F. and Miller, M., 2000. Two new (a, d)-antimagic graph labelings. In Proceedings of Eleventh Australasian Workshop on Combinatorial Algorithms (Vol. 11, pp. 179-189).[Google Scholor]

[16] Sudarsana, I.W., Ismaimuza, D., Baskoro, E.T. and Assiyatun, H., 2005. On super (a, d)-edge antimagic total labeling of disconnected graphs. Journal of Combinatorial Mathematics and Combinatorial Computing, 55, pp.149-158.[Google Scholor]

[17] Gutiérrez, A. and Lladó, A., 2005. Magic coverings. Journal of Combinatorial Mathematics and Combinatorial Computing, 55, 43-56.[Google Scholor]

[18] Muntaner-Batle, F.A., 2001. Special Super Edge Magic-Labelings of Bipartite Graphs. Journal of Combinatorial Mathematics and Combinatorial Computing, 39, pp.107-120.[Google Scholor]

[19] Kumar, S. S., and Marimuthu, G. A H-V-super-strong magic decomposition of complete bipartite graphs, communicated.

[20] Inayah, N., Lladó, A. and Moragas, J., 2012. Magic and antimagic H-decompositions. Discrete Mathematics, 312(7), pp.1367-1371.[Google Scholor]

[21] Liang, Z., 2012. Cycle-supermagic decompositions of complete multipartite graphs. Discrete Mathematics, 312(22), pp.3342-3348.[Google Scholor]

[22] Egawa, Y., Urabe, M., Fukuda, T. and Nagoya, S., 1986. A decomposition of complete bipartite graphs into edge-disjoint subgraphs with star components. Discrete mathematics, 58(1), pp.93-95.[Google Scholor]

[23] Akiyama, J. and Kano, M., 1984. Path factors of a graph, Graphs and Applications, Wiley, Newyork.[Google Scholor]

[24] Marimuthu, G.T. and Kumar, S.S., H-E-super magic decomposition of complete bipartite graphs, communicated.[Google Scholor]

[25] Kumar, S.S. and Marimuthu, G.T., 2015. HV-super magic decomposition of complete bipartite graphs. Communications of the Korean Mathematical Society, 30(3), pp.313-325.[Google Scholor]

Contents

Ars Combinatoria

$H$ - $V$ -Super-Strong- $(a, d)$ -antimagic decomposition of complete bipartite graphs

Abstract

1. Introduction

2. Main Results

3. Conclusion

Conflict of Interest

References:

Information

Guidelines

CP Initiatives

Follow CP

Contents

Ars Combinatoria

H-V-Super-Strong-(a,d)-antimagic decomposition of complete bipartite graphs

Abstract

1. Introduction

2. Main Results

3. Conclusion

Conflict of Interest

References:

Information

Guidelines

CP Initiatives

Follow CP

$H$ - $V$ -Super-Strong- $(a, d)$ -antimagic decomposition of complete bipartite graphs