Pelabelan Super (a,d)- {H} Antimagic Total Dekomposisi pada Shakel dari Graf Kipas Konektif

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 $H$.}