Independence and the Havel-Hakimi residue

Favaron et al. (1991) have obtained a proof of a conjecture of Fajtlowicz’ computer program Graffiti that for every graph G the number of zeroes left after fully reducing the degree sequence as in the Havel-Hakimi Theorem is at most the independence number of G. In this paper we present a simplified version of the proof of Graffiti’s conjecture, and we find how the residue relates to a natural greedy algorithm for constructing large independent sets in G. The Havel-Hakimi Theorem [S, 63 recursively characterizes the sorted integer sequences (d)=(dI>d2>... > d, 20) which arise as the degree sequences of simple graphs. It states that (d) is graphically realizable if and only if the derived sequence L1 (d) is graphically realizable, where Li(d) denotes the sorted sequence obtained after dropping the term di and decreasing the di largest other terms by one each. Hence, (d) is graphical if and only if the repeated application of operation L1 leads to a sequence of zeroes. For example, (633333331)+(32222221)+(222111 l)+...+(OOO), so (633333331) is graphically realizable. Fajtlowicz [l] proposed considering the number of zeroes left when a graphically realizable sequence is reduced.This is called the residue, denoted R= R(d), e.g., R(633333331) = 3. His computer program ‘Graffiti’ checked many examples and found that in every case M. 2 R, where tl= LX(G) is the size of the largest set of independent (i.e., Correspondence to: Jerrold R. Griggs, Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA. *Supported in part by NSA/MSP Grant MDA90-H-4028. **Supported in part by grants AF89-0271 and NSF DMS-8606225. 0012-365X/94/$07.00