The deficiency of a hypergraph

In this chapter, we introduce the completely new technique of the deficiency of a hypergraph. Using this concept of deficiency, we prove here a key theorem that we will need in establishing the transversal number of a linear 4-uniform hypergraph, including determining the Tuza constant q₄ in the next chapter. We begin this chapter by defining a few so-called special hypergraphs. We then introduce the concept of the deficiency of a set of hypergraphs. Thereafter, we prove a key theorem that we will need in establishing the transversal number of a linear 4-uniform hypergraph. Throughout this section, we let L_{4, 3} be the class of all 4-uniform, linear hypergraphs with maximum degree at most 3. Thus, L_{4, 3} is a subclass of the class L₄ of all 4-uniform, linear hypergraphs.