Factorizations of Complete Graphs into $[R,S,T,2)]$-Caterpillars of Diameter 5

Abstract

A tree $R$ such that after deleting all leaves we obtain a path $P$ is called a \emph{caterpillar}. The path $P$ is called the \emph{spine} of the caterpillar $R$. If the spine has length 3 and $R$ on $2n$ vertices contains vertices of degrees $r$, $s$, $t$, $2$, where $2<r,s,t<n$, then we say that $R$ is an $(r,s,t,2)$-\emph{caterpillar} of diameter 5. We completely characterize $(r,s,t,2)$-caterpillars of diameter 5 on $4k+2$ vertices that factorize the complete graph $K_{4k+2}$.