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An \(H\)-(a,d)-antimagic labeling in a \(H\)-decomposable graph \(G\) is a bijection \(f: V(G)\cup E(G)\rightarrow {\{1,2,…,p+q\}}\) such that \(\sum f(H_1),\sum f(H_2),\cdots, \sum f(H_h)\) forms an arithmetic progression with difference \(d\) and first element \(a\). \(f\) is said to be \(H\)-\(V\)-super-\((a,d)\)-antimagic if \(f(V(G))={\{1,2,…,p\}}\). 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 \(f(U(G))={\{1,2,…,m\}}\) and \(f(W(G))={\{m+1,m+2,…,(m+n=p)\}}\). 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.
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
\(1,2,\cdots,q\) 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\(\check{a}\)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 \({1,2,…,p+q}\) is called a
vertex-antimagic total labeling of \(G\) if the weights of vertices \(\{w_f (x)=f(x)+\sum_{xy \in E(G)} f(xy)\),
\(x \in V(G)\)}, are pairwise distinct.
\(f\) is called an \((a,d)\)-vertex-antimagic total labeling of
\(G\) if the set of vertex weights
\(\{w_f (x) | x \in
V(G)\}=\{a,a+d,\cdots,a+(p-1)d\}\) for some integers \(a\) and \(d\). \(f\)
is said to be super-\((a,d)\)-vertex-antimagic labeling if \(f(V(G))={\{1,2,…,p\}}\). 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 \({1,2,…,p+q}\) is called an \((a,d)\)-edge-antimagic total labeling of
\(G\) if the edge weights \(\{w(uv)=f(u)+f(v)+f(uv)\), \(uv \in E(G)\)}, 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 \(f(V(G))={\{1,2,…,p\}}\). 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 \(H_1,H_2,…,H_h\) 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 \((H_1,H_2,\cdots ,H_h)\) covering. If every
\(H_i\) is isomorphic to a given graph
\(H\), then \(G\) admits an \(H\)-covering. A family of subgraphs \(H_1,H_2,\cdots,H_h\) 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 \(G=H_1\oplus H_2 \oplus\cdots \oplus
H_h\) and \(G\) is said to be
\(H\)-decomposable.
The notion of \(H\)-super magic
labeling was first introduced and studied by Guti\(\acute{e}\)rrez and Llad\(\acute{o}\)[17] in 2005. They proved that some classes of
connected graphs are \(H\)-super magic.
Suppose \(G\) is \(H\)-decomposable. A total labeling \(f :V(G)\cup E(G)\to {\{1,2,\cdots,p+q\}}\)
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 \(f(V(G))={\{1,2,\cdots,p\}}\). 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 \(f(E(G))={\{1,2,\cdots,q\}}\). 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
\({\{1,2,…,m\}}\) and the elements of
\(V_2\) receive labels \({\{m+1,m+2,…,m+n\}}\). 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 \(f(U(G))={\{1,2,…,m\}}\) and \(f(W(G))={\{m+1,m+2,…,(m+n=p)\}}\). 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 \(f(U(G))={\{q+1,q+2,…,q+m\}}\) and \(f(W(G))=\{q+m+1,q+m+2,…,\\(q+m+n=qp)\}\).
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 \(f :V(G)\cup E(G)\to \{1,2,\cdots, \\
p+q\}\) is called an \(H\)-antimagic labeling of \(G\) if \(\sum
f(H_1),\sum f(H_2), \cdots, \sum f(H_h)\) 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 \(f(V(G))={\{1,2,…,p\}}\). 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 \(f(U(G))={\{1,2,…,m\}}\) and \(f(W(G))={\{m+1,m+2,…,(m+n=p)\}}\). 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 \(f(E(G))={\{1,2,…,q\}}\). \(f\) is said to be \(H\)–\(E\)-super-strong-\((a,d)\)-antimagic labeling if \(f(U(G))={\{q+1,q+2,…,q+m\}}\) and \(f(W(G))=\{q+m+1,q+m+2,…,(q+m+n=qp)\}\). 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 \(m\geq1\). A subgraph \(F\) of a graph \(G\) is called a \(n\)-star-subgraph if \(F \cong K_{1,n}\) 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.
In this section, we consider the graphs \(G \cong K_{m,n}\) and \(H \cong K_{1,n}\), where \(n\geq1\) and both \(m\) and \(n\) are even. Clearly \(p=m+n\) and \(q=mn\).
Theorem 1. Suppose \({\{H_1,H_2,\cdots ,H_m\}}\) 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 \(a=1+\frac{n^2(m+3)+2n(2m+1)}{2}\) and \(d=1\).
Proof. Let \(U={\{u_1,u_2,\cdots
,u_m\}}\) and \(V={\{v_1,v_2,\cdots
,v_n\}}\) be two stable sets of \(G\). Let \({\{H_1,H_2,\cdots ,H_m\}}\) be a \(n\)-star decomposition of \(G\) with both \(m\) and \(n\) are even, where each \(H_i\) is isomorphic to \(H\), such that \(V(H_i)={\{u_i,v_1,v_2,\cdots ,v_n\}}\) and
\(E(H_i)={\{u_iv_1,u_iv_2, \cdots
,u_iv_n\}}\), for all \(1\leq i \leq
m\). Define a total labeling \(f
:V(G)\cup E(G) \rightarrow {\{1,2, \cdots ,p+q\}}\) 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 \(1\).
| \(f\) | \(v_1\) | \(v_2\) | \(…\) | \(v_{n-1}\) | \(v_n\) |
|---|---|---|---|---|---|
| \(u_1\) | \((m+n)\) | \((2m+n)\) | \(…\) | \((m+n)\) | \((m+n)\) |
| \(\) | \(+m\) | \(+1\) | \(\) | \(+((n-1)m)\) | \(+((n-1)m+1)\) |
| \(u_2\) | \((m+n)+\) | \((2m+n)\) | \(…\) | \((m+n)\) | \((m+n)\) |
| \(\) | \((m-1)\) | \(+2\) | \(\) | \(+((n-1)m-1)\) | \(+((n-1)m+2)\) |
| \(u_3\) | \((m+n)+\) | \((2m+n)\) | \(…\) | \((m+n)\) | \((m+n)\) |
| \(\) | \((m-2)\) | \(+3\) | \(\) | \(((n-1)m-2)\) | \(+((n-1)m+3)\) |
| \(\vdots\) | \(…\) | \(…\) | \(…\) | \(…\) | \(…\) |
| \(u_k\) | \((m+n)+\) | \((2m+n)\) | \(…\) | \((m+n)+((n-2)m)\) | \((m+n)+(n-1)m\) |
| \(\) | \((m-(k-1))\) | \(+k\) | \(\) | \(+(m-(k-1))\) | \(+k\) |
| \(\vdots\) | \(…\) | \(…\) | \(…\) | \(…\) | \(…\) |
| \(u_{m-1}\) | \((m+n)+\) | \((2m+n)\) | \(…\) | \((m+n)\) | \((m+n)\) |
| \(\) | \(2\) | \(+(m-1)\) | \(\) | \(+((n-2)m+2)\) | \(+(mn-1)\) |
| \(u_m\) | \((m+n)+\) | \((2m+n)\) | \(…\) | \((m+n)\) | \((m+n)\) |
| \(\) | \(1\) | \(+m\) | \(\) | \(+((n-2)m+1)\) | \(+mn\) |
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 \[\begin{aligned}
\sum f(H_k) &=& f(u_k)+\sum_{i=1}^n f(v_i)+\sum_{i=1}^n f(u_k
v_i )= k+\sum_{i=1}^n (m+i)+\sum_{i=1}^n f(u_k v_i ) \nonumber.
\end{aligned}\] Now, \[\begin{aligned}
\sum_{i=1}^n f(v_i) &=& (m+1)+(m+2)+\cdots+(m+n) \nonumber \\
&=& mn+(1+2+\cdots+n) = mn+\frac{n(n+1)}{2}
\nonumber.
\end{aligned}\] Also \[\begin{aligned}
\sum_{i=1}^n f(u_k v_i) &=&
((m+n)+(m-(k-1)))+((m+n)+(m+k))+\cdots \nonumber \\
&\ \
& +((m+n)+(n-2)m+(m-(k-1)))+((m+n)+(n-1)m+k) \nonumber \\
&=&
((2m+n)-(k-1))+((2m+n)+k)+((4m+n)-(k-1))+ \nonumber \\
&\ \ &
((4m+n)+k)+\cdots+(((n)m+n)-(k-1))+(((n)m+n)+k) \nonumber \\
&=& 2((2m+n)+(4m+n)+\cdots
+(nm+n))+\frac{n}{2}(1) \nonumber \\
&=&
2((2m+2n+\cdots+nm)+\frac{n(n)}{2})+\frac{n}{2} \nonumber \\
&=&
4m(1+2+\cdots+\frac{n}{2})+\frac{2n^2+n}{2} =
4m(\frac{n(n+2)}{8})+\frac{2n^2+n}{2} \nonumber \\
&=& \frac{mn^2+2mn+2n^2+n}{2} =
\frac{n^2(m+2)+n(2m+1)}{2} \nonumber.
\end{aligned}\]
Hence \[\begin{aligned}
\sum_{i=1}^n f(u_k v_i) &=& \frac{n^2(m+2)+n(2m+1)}{2}
\nonumber.
\end{aligned}\] and is constant for all \(1 \leq k \leq m\).
Using the above values, we get \[\begin{aligned}
\sum f(H_k) &=& k+mn+\frac{n(n+1)}{2}+\frac{n^2(m+2)+n(2m+1)}{2}
\nonumber \\
&=&
k+\frac{2mn+n^2+n+n^2(m+2)+n(2m+1)}{2} \nonumber \\
&=& k+\frac{n^2(m+3)+2n(2m+1)}{2}
\nonumber.
\end{aligned}\] for all \(1 \leq k \leq
m\). So, \(\{\sum f(H_1),\sum
f(H_2),\cdots,\sum f(H_m)=a,a+d,\cdots,a+(m-1)d\}\) forms an
arithmetic progression with \(a=
(1+\frac{n^2(m+3)+2n(2m+1)}{2})\) 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 \(2\).
| \(f\) | \(v_1\) | \(v_2\) | \(…\) | \(v_{n-1}\) | \(v_n\) |
|---|---|---|---|---|---|
| \(u_1\) | \(3n\) | \(3n+1\) | \(…\) | \((n+1)n\) | \((n+1)n+1\) |
| \(u_2\) | \(3n-1\) | \(3n+2\) | \(…\) | \((n+1)n-1\) | \((n+1)n+2\) |
| \(u_3\) | \(3n-2\) | \(3n+3\) | \(…\) | \((n+1)n-2\) | \((n+1)n+3\) |
| \(\vdots\) | \(…\) | \(…\) | \(…\) | \(…\) | \(…\) |
| \(u_k\) | \(3n-(k-1)\) | \(3n+k\) | \(…\) | \((n+1)n-(k-1)\) | \((n+1)n+k\) |
| \(\vdots\) | \(…\) | \(…\) | \(…\) | \(…\) | \(…\) |
| \(u_{n-1}\) | \(2n+2\) | \(4n-1\) | \(…\) | \(n(n)+2\) | \((n+2)n-1\) |
| \(u_n\) | \(2n+1\) | \(4n\) | \(…\) | \(n(n)+1\) | \((n+2)n\) |
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 \[\begin{aligned}
\sum f(H_k) &=& f(u_k)+\sum_{i=1}^n f(v_i)+\sum_{i=1}^n f(u_k
v_i ) = k+\sum_{i=1}^n (n+i)+\sum_{i=1}^n f(u_k v_i ) \nonumber.
\end{aligned}\] Now, \[\begin{aligned}
\sum_{i=1}^n f(v_i) &=& (n+1)+(n+2)+\cdots+(n+n)=
(n)n+(1+2+\cdots+n) \nonumber \\
&=& (n)n+\frac{n(n+1)}{2} \nonumber.
\end{aligned}\] Also \[\begin{aligned}
\sum_{i=1}^n f(u_k v_i) &=&
(3n-(k-1))+(3n+k)+(5n-(k-1))+(5n+k)+\cdots \nonumber \\
&\ \ & +((n+1)n-(k-1))+((n+1)n+k)
\nonumber \\
&=&
(3n+1)+3n+(5n+1)+5n+\cdots+((n+1)n+1)+(n+1)n \nonumber \\
&=& 2(3n+5n+\cdots
+(n+1)n)+\frac{n}{2}(1) \nonumber \\
&=& 2n(3+5+\cdots
+(n+1))+\frac{n}{2} \nonumber \\
&=&
2n((1+2+3+\cdots+(n+1))-(2+4+6+\cdots+n)-1)+\frac{n}{2} \nonumber \\
&=&
2n(\frac{(n+1)(n+2)}{2}-2\frac{(\frac{n}{2})(\frac{n+1}{2})}{2}-1)+\frac{n}{2}
\nonumber \\
&=&
2(\frac{n^2+3n+2}{2}-\frac{(n^2+2n)}{4}-1)+\frac{n}{2} \nonumber \\
&=&
2n(\frac{2n^2+6n+4-n^2-2n-4}{4}+\frac{n}{2} = \frac{n(n^2+4n+n)}{2}
\nonumber \\
&=& \frac{n^3+2n^2+2n^2+n}{2} =
\frac{n^2(n+2)+(n(2n+1)}{2} \nonumber.
\end{aligned}\] Hence \[\begin{aligned}
\sum_{i=1}^n f(u_k v_i) &=& \frac{n^2(n+2)+n(2n+1)}{2}
\nonumber.
\end{aligned}\] and is constant for all \(1 \leq k \leq n\).
Using the above values, we get \[\begin{aligned}
\sum f(H_k) &=&
k+(n)n+\frac{n(n+1)}{2}+\frac{n^2(n+2)+n(2n+1)}{2} \nonumber \\
&=&
k+\frac{2(n)n+n^2+n+n^2(n+2)+n(2n+1)}{2} \nonumber \\
&=& k+\frac{n^2(n+3)+2n(2n+1)}{2}
\nonumber.
\end{aligned}\] for all \(1 \leq k \leq
n\). So, \(\{\sum f(H_1),\sum
f(H_2),\cdots,\sum f(H_n)=a,a+d,\cdots,a+(n-1)d\}\) forms an
arithmetic progression with \(a=
(1+\frac{n^2(n+3)+2n(2n+1)}{2})\) 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 \(G\cong K_{m,n}\) 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 \(\sum f(E(H_j))=\frac{n^2(m+2)+n(2m+1)}{2}\).
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 \[\begin{aligned} \sum f(E(H_j)) &=& \sum_{i=1}^n f(u_jv_i)= \frac{n^2(m+2)+n(2m+1)}{2} \end{aligned}\] 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 \(\sum f(E(H_j))=\frac{n^2(m+2)+n(2m+1)}{2}\). ◻
Theorem 3. If a non-trivial \(H\)-decomposable graph \(G\cong K_{m,n}\) 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
\(\{\sum f(V(H_1)),\sum f(V(H_2)),\cdots,
\sum f(V(H_m))\}=\{a,a+d,\cdots,a+(m-1)d\}\) with \(a=(mn+1)+\frac{n(n+1)}{2}\) 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 \[\begin{aligned} \sum f(V(H_j)) &=& f(u_j)+\sum_{i=1}^n f(v_i) = j+\sum_{i=1}^n (m+i) = j+((m+1)+(m+2)+\cdots+(m+n)) \nonumber \\ &=& j+mn+\frac{n(n+1)}{2} \nonumber. \end{aligned}\] which is true for all \(1 \leq j \leq m\). Thus \(\{\sum f(V(H_1)),\sum f(V(H_2)),\cdots, \\ \sum f(V(H_m))\}=\{a,a+d,\cdots,a+(m-1)d\}\) with \(a=(mn+1)+\frac{n(n+1)}{2}\) and \(d=1\). ◻
Theorem 4. Let \(G\cong K_{m,n}\) 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 \(\{1,2,\cdots,p\}\) with \(g(U(G))=\{1,2,\cdots,m\}\) and \(g(W(G))=\{(m+1),(m+2),\cdots,(m+n=p)\}\) 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 \(\sum f(E(H_j))=\frac{n^2(m+2)+n(2m+1)}{2}\).
Proof. Suppose \(G\cong
K_{m.n}\) 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 \(\{1,2,\cdots,p\}\) with \(g(U(G))=\{1,2,\cdots,m\}\) and \(g(W(G))=\{(m+1),(m+2),\cdots,(m+n=p)\}\).
Assume that \(\sum f(E(H_j))\) is
constant for all \(1 \leq j \leq m\)
and it is given by \(\sum
f(E(H_j))=\frac{n^2(m+2)+n(2m+1)}{2}\). Define \(f: V(G)\cup E(G)\rightarrow
{\{1,2,…,p+q\}}\) 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 \[\begin{aligned}
\sum f(V(H_j)) &=& f(u_j)+\sum_{i=1}^n f(v_i)= j+\sum_{i=1}^n
(m+i) = j+((m+1)+(m+2)+\cdots+(m+n)) \nonumber \\
&=& j+mn+\frac{n(n+1)}{2} \nonumber.
\end{aligned}\] which is true for all \(1 \leq j \leq m\). So, we have \(\{\sum f(V(H_1)),\sum f(V(H_2)),\cdots, \sum
f(V(H_m))\}=\{a,a+d,\cdots,a+(m-1)d\}\) with \(a=(mn+1)+\frac{n(n+1)}{2}\) and \(d=1\). Hence, \[\begin{aligned}
\sum f(H_j)&=& \sum f(V(H_j)) +\sum f(E(H_j))=
(j+mn+\frac{n(n+1)}{2})+ (\frac{n^2(m+2)+n(2m+1)}{2}) \nonumber \\
&=& j+\frac{2mn+n^2+n+n^2(m+2)+n(2m+1)}{2}
= j+\frac{n^2(m+3)+2n(2m+1)}{2} \nonumber.
\end{aligned}\] 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 \(a=1+\frac{n^2(m+3)+2n(2m+1)}{2}\) 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 \(\sum
f(E(H_j))=\frac{n^2(m+2)+n(2m+1)}{2}\). ◻
In this paper, we studied the \(H\)–\(V\)-super-strong-\((a,d)\)-antimagic decomposition of \(K_{m,n}\) with \(n\geq1\) and both \(m\) and \(n\) are even.
The author declares no conflict of interests.