On the Domination Order among Elimination Sequences

TL;DR

This paper proves that the elimination sequence from the Havel-Hakimi algorithm dominates all others for degree sequences, confirming a conjecture that the number of zeros in any elimination process is bounded by the residue.

Contribution

It establishes that the Havel-Hakimi elimination sequence dominates all other elimination sequences for degree sequences, confirming Barrus's conjecture.

Findings

Havel-Hakimi elimination sequence dominates others

Confirms Barrus's conjecture on zeros bounded by residue

Provides a unified framework for elimination sequences

Abstract

In 1991, it was shown by Favaron, Mah\'eo, and Sacl\'e that the residue, which is defined as the number of zeros remaining when the Havel-Hakimi algorithm is applied to a degree sequence, yields a lower bound on the independence number of any graph realising the sequence. In 1996, Triesch simplified and generalised the result by introducing elimination sequences. It was proved in 1973 by Kleitman and Wang that for any graphic sequence all elimination algorithms, i.e. laying-off vertices in any order, preserve that the sequence is graphic and terminate in a sequence of zeros. We now prove that for any degree sequence, the elimination sequence derived from the Havel-Havel algorithm dominates all other elimination sequences. Our result implies a conjecture posed by Michael Barrus in 2010: When iteratively laying off degrees from a graphic sequence until only a list of zeros remains, the number of zeros is at most the residue of this sequence.