Two short proofs of the bounded case of S.B. Rao's degree sequence conjecture

TL;DR

This paper presents two concise proofs for the bounded case of S. B. Rao's degree sequence conjecture, leveraging properties of integer sequences and their graphical nature, contributing to the understanding of well-quasi-ordering in graph theory.

Contribution

It provides two novel, short proofs of the bounded case of Rao's conjecture, simplifying previous approaches and highlighting key properties of integer sequences.

Findings

Both proofs confirm the bounded case of Rao's conjecture.

The proofs utilize the relationship between sequence length and maximum term.

They demonstrate that sequences with many entries relative to their maximum are necessarily graphic.

Abstract

S. B. Rao conjectured that graphic sequences are well-quasi-ordered under an inclusion based on induced subgraphs. This conjecture has now been settled completely by M. Chudnovsky and P. Seymour. One part of the proof proves the result for the bounded case, a result proved independently by C. J. Altomare. We give two short proofs of the bounded case of S. B. Rao's conjecture. Both the proofs use the fact that if the number of entries in an integer sequence (with even sum) is much larger than its highest term, then it is necessarily graphic.