Decomposition of complete graphs into unicyclic graphs with eight edges

Abstract

Let \( G \) be a tripartite unicyclic graph with eight edges that either (i) contains a triangle or heptagon, or (ii) contains a pentagon and is disconnected. We prove that \( G \) decomposes the complete graph \( K_n \) whenever the necessary conditions are satisfied. We combine this result with other known results to prove that every unicyclic graph with eight edges other than \( C_8 \) decomposes \( K_n \) if and only if \( n \equiv 0,1 \pmod{16} \).