All 2-regular graphs with uniform odd components admit ρ-labellings

Let G be a graph of size n with vertex set V (G) and edge set E(G). A ρlabeling of G is a one-to-one function h : V (G) → {0, 1, . . . , 2n} such that {min{|h(u)−h(v)|, 2n+1−|h(u)−h(v)|} : {u, v} ∈ E(G)} = {1, 2, . . . , n}. Such a labeling of G yields a cyclicG-decomposition ofK2n+1. It is known that 2-regular bipartite graphs, the vertex-disjoint union of C3’s, and the vertex-disjoint union of C5’s all admit ρ-labelings. We show that for any odd n ≥ 7, the vertex-disjoint union of any number of Cn’s admits a ρ-labeling.