Factorizations of complete graphs into [n, r, s, 2]-caterpillars of diameter 5 with maximum end

Abstract

A tree R such that after deleting all leaves we obtain a path P is called a caterpillar. The path P is called the spine of the caterpillar R. If the spine has length 3 and R on 2n vertices contains vertices of degrees n, r, s, 2, where r, s>2, and if a vertex of degree n is an endvertex of the spine then we say that R is an [n, r, s, 2] -caterpillar with maximum end of diameter 5. We completely characterize [2k + 1, r, s, 2]-caterpillars of order 4k + 2 and diameter 5 with maximum end that factorize the complete graph K4k+2.