Existence of k-edge connected ordinary graphs with prescribed degrees

Abstract

An ordinary graph G is a set of objects called nodes and a family of unord e re d pai rs of. th e nodes called edges. The degree of a node in G is the number of edges in G whi ch co ntain it. G is called connected if it is not the union of two di sjoint none mpty subgraphs. A graph H is called k-edge connected if deletin g any fewer than k edges from H leaves a co nn ected graph. It is proved that th ere exists a k-edge connected gr aph H fo r k > 1 with prescribed int.ege r degrees d; if and only if th ere exists an ordinary graph with th ese degrees a nd all d; "" k. The re ex ist.s a l-co nn ec ted (i.e., co nnec ted) ordinary graph with presc ribed pos itive integer degrees d; if and only if the re ex ists an ordinary graph " with these degrees and L d; "" 2(n-l). An ordinary graph G is a finite set of objects and a family of two-m e mber sub se ts of th e objects. The objects are called the nodes of G and the pairs are called the edges of G. An edge and a nod e are said to meet if one contains the oth er. The degree of a node in G is the number of edges in G which it meets. A cut of graph G, de noted by (5,5), is a partition of the nodes of G into_ two none mpty s ubsets 5 and S.