The principal Erd\H{o}s--Gallai differences of a degree sequence

TL;DR

This paper studies the differences in the Erd ext{"o}s--Gallai inequalities for degree sequences, revealing their invariance under graph complementation and their monotonicity properties, with implications for characterizing special graph classes.

Contribution

It introduces new invariance and monotonicity properties of the Erd ext{"o}s--Gallai differences, extending understanding of degree sequence characterizations.

Findings

Last and maximum Erd ext{"o}s--Gallai differences are preserved under graph complementation.

Differences are monotonic under majorization and Rao's order.

Survey of applications in split and threshold graphs.

Abstract

The Erd\H{o}s--Gallai criteria for recognizing degree sequences of simple graphs involve a system of inequalities. Given a fixed degree sequence, we consider the list of differences of the two sides of these inequalities. These differences have appeared in varying contexts, including characterizations of the split and threshold graphs, and we survey their uses here. Then, enlarging upon properties of these graph families, we show that both the last term and the maximum term of the principal Erd\H{o}s--Gallai differences of a degree sequence are preserved under graph complementation and are monotonic under the majorization order and Rao's order on degree sequences.