AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 53 (2012), Pages 207–219 All 2-regular graphs with uniform odd components admit ρ-labelings D. I. Gannon Department of Mathematics University of Illinois Urbana, Illinois 61801 U.S.A. danitoh@aol.com S. I. El-Zanati ∗ 4520 Mathematics Department Illinois State University Normal, Illinois 61790–4520 U.S.A. saad@ilstu.edu Abstract 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 K 2n+1 . It is known that 2-regular bipartite graphs, the vertex-disjoint union of C 3 ’s, and the vertex-disjoint union of C 5 ’s all admit ρ-labelings. We show that for any odd n ≥ 7, the vertex-disjoint union of any number of C n ’s admits a ρ-labeling. 1 Introduction If a and b are integers we denote {a, a +1,...,b} by [a, b] (if a>b,[a, b]= ∅). Let N denote the set of nonnegative integers and Z n the group of integers modulo n. For a graph G, let V (G) and E(G) denote the vertex set of G and the edge set of G, respectively. Let rG denote the vertex-disjoint union of r copies of G. * Research supported by National Science Foundation Grant No. A0649210