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

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 \( K_{4k+2} \).