Decomposition of complete graphs into unicyclic bipartite graphs with eight edges

Abstract

We introduce a variation of $\sigma$-labeling to prove that every disconnected unicyclic bipartite graph with eight edges decomposes the complete graph $K_n$ whenever the necessary conditions are satisfied. We combine this result with known results in the connected case to prove that every unicyclic bipartite graph with eight edges other than $C_8$ decomposes $K_n$ if and only if $n\equiv 0,1(\mathrm{mod}16)$ and $n>16$.