[a, b]-Factorizations

Abstract

Let G be a graph, and g, f : V (G) → ℤ be functions such that g(x) = f(x) for all x ∈ V (G). If the set of edges of G can be decomposed into disjoint subsets $E(G) = {F_1}\cup{F_2}\cup\cdots\cup{F_n}$ so that every Fi induces a (g, f)-factor of G, then we say that G is (g, f)- factorable, and the above decomposition is called a (g, f)-factorization of G.We often regard an edge set F of a graph as its spanning subgraph with edge set F. As a special case of (g, f)-factorization, we can define 1-factorization, k -regular factorization, [a, b]-factorization and f -factorization. In this chapter, we mainly investigate [a, b]-factorizations of graphs. We begin with some basic results on factorizations of special graphs.