Antimagicness of $mC_n$ -Path and Its Disjoint Union

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.