Some Research On The Factors Of Graphs, (g,f)-k-deleted Graphs And (g,f)-k-covered Graphs

All graphs under consideration are undirected and finite. Multiple edges are allowed but loops are not without special notes.Let G be a graph with vertex set V(G) and edge set E(G). For a vertex x @ V(G), we write dG (x) for the degree of x in G and NG (x) for the neighborhood of x in G. We write S(G) for the minimum degree of G.Let g(x) and f(x) be integer-valued functions defined on V(G) with g(x) ≤ f(x) for each vertex x e V(G). A graph G is called a (g, /) - graph ifg(x) ≤ dG (x) ≤ f(x) for each vertex x e V{G) and a (g,f) -factor of a graph is a spanning (g,f)- sub-graph of G. Ifg(x) = a, f(x} = b for all x@V(G), we call a (g,f)-factor to be a [a,b]-factor; if g(x) = f(x) = k for all x@V(G),a(g, f) - factor of G is called a k - factor. Similarly, we can define [a, b] - graph and k- regular graph.If for any edge e of G there is a (g,f) - factor containing e, then G is calledcovered graph. A graph G is called a deleted graph if every edge does not belong to a factor. Generalizing these concepts, we have the following ones. Suppose k is apositive integer. If for every k edges of G there is a (g, f) - factor containing these k edges, then G is called (g, f)- k - covered graph. On the contrary, a graph G iscalled a (g,f) - k - deleted graph if every k edges don't belong to a (g, f) - factor.The problem of factors in graphs is one of the main problems in the research of graph theory. So far, many results on the existence of factors in graphs have been gained. In this paper, we mainly consider the following problems: the relationship between the existence of a k - factor in graph and the minimum degree, the existenceof (g,f)-k-deleted graph and (g, f)- k - covered graph (k = 2,3) and so on.The full paper is separated into five sections.In the first section, we give a brief introduction about the basic concepts terms and signs which are involved in this paper. At the same time, we summarize some important theorems and lemmas which are useful in the later proofs.In the second section, we give a sufficient condition for the existence of a k - factor in a graph in terms of the minimum degree.In the third section, some necessary and sufficient conditions for the existence of(g,f)-k-deletedgraph (k = 2,3).In the fourth section, on the contrary we give some necessary and sufficient conditions for the existence of (g,f) - k - covered graph (k = 2,3)In the last part, we put some new problems and conjectures.