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

Dalibor Froncek (University of Minnesota Duluth)
O'Neill Kingston (Iowa State University)

Indonesian Journal of Combinatorics
Vol 3, No 1 (2019)

A G-decomposition of the complete graph Kn is a family of pairwise edge disjoint subgraphs of Kn, all isomorphic to G, such that every edge of Kn 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 Kn whenever the necessary conditions are satisfied.