IEEE Access (Jan 2019)
Antimagicness of <inline-formula> <tex-math notation="LaTeX">$mC_n$ </tex-math></inline-formula>-Path and Its Disjoint Union
Abstract
Let $M=(V(M),E(M))$ be a simple graph with finite vertices and edges. An $N$ -covering of $M$ is a family $\{N_{1}, N_{2}, {\dots },N_{\alpha }\}$ of subgraphs of $M$ isomorphic to $N$ such that every edge in $E(M)$ belongs to $N_{l}$ , for some $l,~l\in \{1, 2, {\dots }, \alpha \}$ . Such a graph is a $(c,d)$ - $N$ -antimagic if $\exists $ a bijection $\psi:V_{M}\cup E_{M} \to \{1,2, {\dots }, |V_{M}|+|E_{M}| \}$ such that for all $N_{l}\cong N$ , $\{wt_{\psi }(N_{l})\}=\{c, c+d, {\dots }, c+(\alpha -1)d\}$ . For $\psi (V(M))= \{ 1,2,3, {\dots },|V(M)|\}$ , the labeling $\psi $ would be super $(c,d)-N$ -antimagic and for $d=0$ it would be $N$ -supermagic. In this manuscript, we investigated that $mC_{n}$ -path has super $(c,d)$ - $C_{n}$ -antimagic labeling for differences $d\in \{0,1, {\dots }, 5\}$ and extend the result for $C_{n}$ -supermagic labeling of disjoint union of isomorphic copies of $mC_{n}$ -path.
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