On $\rho$-labeling Up to Ten Vertex-disjoint $C_{4x+1}$

Abstract

Let $G$ be a graph of size $n$ with vertex set $V(G)$ and edge set $E(G)$. A $\rho$-$labeling$ 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.