Seven Criteria for Integer equences Graphic Ill e1ng ABSTRACT Gerard Sierksma DEPARTMENT OF ECONOMETRICS, SECTION OPERATIONS RESEARCH UNIVERSITY OF GRONINGEN, NETHERLANDS Han Hoogeveen CENTRE FOR MATHEMATICS AND COMPUTER SCIENCE AMSTERDAM, NETHERLANDS Seven criteria for integer sequences being graphic are listed. Being graphic means that there is a simple graph with the given integer sequence as de- gree sequence. One of the criteria leads to a new and constructive proof of the well-known criterion of Erdos-Gallai. 1. INTRODUCTION Let (di, ... , dn) be a nonincreasing sequence of positive integers with even sum. The sequence (di, ... , dn) is called graphic iff there is a simple graph (without loops and multiple edges) that has (di, ... dn) as degree sequence. In this paper seven criteria for such an integer sequence being graphic are listed; one of these is well known and due to Erdos and Gallai [5]. Proofs of the Erdos-Gallai Criterion can be found in Berge [1] and in Harary [10]. Harary's proof is rather lengthy and Berge's proof uses flows in networks. Recently, Choudum [4] has given a different proof which, in our opinion, is not very appealing either. Using the discovered Hasselbarth Crite- rion we are able to give a new and elegant proof. 2. THE SEVEN CRITERIA In Theorem 1 it will be shown that the following conditions (A)-(G) are all equivalent to "(d 1 , •• • dn) is graphic." Journal of Graph Theory, Vol. 15, No. 2, 223-231 (1991) © 1991 John Wiley & Sons, Inc. CCC 0364-9024/91/020223-09$04.00