Decomposition of complete graphs into connected unicyclic graphs with eight edges and pentagon

Abstract

A G-decomposition of the complete graph K_n is a family of pairwise edge disjoint subgraphs of K_n, all isomorphic to G, such that every edge of K_n belongs to exactly one copy of G. Using standard decomposition techniques based on ρ-labelings, introduced by Rosa in 1967, and their modifications we show that each of the ten non-isomorphic connected unicyclic graphs with eight edges containing the pentagon decomposes the complete graph K_n whenever the necessary conditions are satisfied.