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 $n\equiv 0,1(\mathrm{mod}16)$.