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On ρ-labeling up to ten vertex-disjoint C4x+1

E. Butzen1, S. I. El-Zanatif H. Jordon1, A. Modica1, R. Schrishuhn1
14520 Mathematics Department Illinois State University Normal, Illinois 61790-4520, U.S.A.

Abstract

Let G be a graph of size n with vertex set V(G) and edge set E(G). A ρ-\emph{labeling} of G is a one-to-one function f:V(G){0,1,,2n} such that {|f(u)f(v)|:{u,v}E(G)}={x1,x2,,xn}, where for each i{1,2,,n} either xi=i or xi=2n+1i. 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 union of up to ten vertex-disjoint C4x+1 admits a ρ-labeling.