Embedding a Restricted Class of Partial $K_4$-Designs

Abstract

Given a partial $K_4$-design $(X,P)$, if $x\in X$ is a vertex which occurs in exactly one block of $P$, then call $x$ a free vertex. In this paper, a technique is described for obtaining a cubic embedding of any partial $K_4$-design with the property that every block in the partial design contains at least two free vertices.