| dc.description.abstract | All graph in this paper are finite, simple
and undirected. By $H'$-covering, we mean every edge in $E(G)$
belongs to at least one subgraph of $G$ isomorphic to a given graph
$H$. A graph $G$ is said to be an $(a, d)$-${\mathcal
{H}}$-antimagic total decomposition if there exist a bijective
function $f: V(G) \cup E(G) \rightarrow \{1, 2,\dots ,|V (G)| +
|E(G)|\}$ such that for all subgraphs $H'$ isomorphic to ${\mathcal
{H}}$, the total ${\mathcal {H}}$-weights $w(H)= \sum_{v\in
V(H')}f(v)+\sum_{e\in E(H')}f(v)$ form an arithmetic sequence $\{a,
a + d, a +2d,...,a+(s - 1)d\}$, where $a$ and $d$ are positive
integers and $s$ is the number of all subgraphs $H'$ isomorphic to
${\mathcal {H}}$. Such a labeling is called super if $f: V(G)
\rightarrow \{1, 2,\dots ,|V (G)|\}$. In this paper, we study the
problem that if a connected graph $G$ is super labelling $(a,
d)-{\mathcal {H}}$- antimagic total decomposition, is the connective
of the graph $G$ super $(a, d)$-${\mathcal {H}}$ - antimagic total
decomposition as well? We will answer this question for the case
when the graph $G$ is a shackle of $SF_4^3$ and ${\mathcal {H}}$=$F_4$ isomorphic to $H$.} | en_US |
