Factorizations of complete graphs into \([n,r,s,2]\)-caterpillars of diameter 5 with maximum center (Q1773844)

A caterpillar of diameter 5 is a tree that arises from the path \(P_6\) by attaching pendant vertices of degree 1 to some or all vertices of degree 2 of the path \(P_6\) (called spine). A \([p_1,p_2,p_3,p_4]\)-caterpillar, where \(p_1\geq p_2\geq p_3\geq p_4\), is a caterpillar of diameter 5 with the spine vertices of degrees \(p_1,p_2,p_3,p_4\) (in any order). The necessary conditions for a spanning \([p_1,p_2,p_3,p_4]\)-caterpillar to factorize a complete graph \(K_m\) are shown: (i) \(m\) must be even, say \(2n\), and (ii) the maximum degree \(p_1\) must be at most \(n\). In the paper only the case of \(n\) odd is considered. A complete characterization of the spanning \([n,p_2,p_3,2]\)-caterpillars with the vertex of maximum degree \(n\) being one of the central vertices of the spine that factorize \(K_{2n}\) is given.