On \(\rho\)-labeling up to ten vertex-disjoint \(C_{4x+1}\)

Let \( G \) be a graph of size \( n \) with vertex set \( V(G) \) and edge set \( E(G) \). A \( \rho \)-\emph{labeling} of \( G \) is a one-to-one function \( f : V(G) \to \{0,1,\dots,2n\} \) such that \( \{|f(u) – f(v)| : \{u,v\} \in E(G)\} = \{x_1,x_2,\dots,x_n\} \), where for each \( i \in \{1,2,\dots,n\} \) either \( x_i = i \) or \( x_i = 2n+1-i \). Such a labeling of \( G \) yields a cyclic \( G \)-decomposition of \( K_{2n+1} \). It is conjectured by El-Zanati and Vanden Eynden that every 2-regular graph \( G \) admits a \( \rho \)-labeling. We show that the union of up to ten vertex-disjoint \( C_{4x+1} \) admits a \( \rho \)-labeling.