On ρ-labeling 2-regular graphs consisting of 5-cycles


Let G be a graph of size n with vertex set V (G) and edge set E(G). A ρ-labeling 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 cyclic G-decomposition of K2n+1. It is conjectured by El-Zanati and Vanden Eynden that every 2-regular graph G admits a ρ-labeling. We show that the vertexdisjoint union of any number of 5-cycles admits a ρ-labeling.


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