Super (a,d)-$H$- Antimagic Total Covering of Chain Graph

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 $(a, d)$-$H$-antimagic total labeling 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 $H$, the total $H$-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 $H$. 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 $(a, d)-H$- antimagic total labeling, is the disjoint union of multiple copies of the graph $G$ super $(a, d)-H$- antimagic total labeling as well? We will answer this question for the case when the graph $G$ is a Chain Graph $K_4 P_n$ and $H'=K_4$ Complete Graph isomorphic to $H$.