Labelled version of the almost bounded case of S. B. Rao's degree sequence conjecture

TL;DR

This paper provides an independent, concise proof for the labelled version of the almost bounded case of S. B. Rao's degree sequence conjecture, where degrees are mostly bounded except for a limited number of vertices.

Contribution

It offers a new, succinct proof for a specific case of Rao's conjecture, expanding understanding of graph degree sequences and their ordering.

Findings

Proof confirms the conjecture in the almost bounded case.

Establishes well-quasi-ordering for the specified class of graph sequences.

Provides a methodological framework for similar proofs in graph theory.

Abstract

S. B. Rao conjectured that graphic sequences are well-quasi-ordered under an inclusion based on induced subgraphs. This conjecture has now been proved by Chudnovsky and Seymour. We give an independent short proof of the labelled version of the almost bounded case of S. B. Rao's conjecture, the case where we have a bound on the degree, but allow a bounded number of vertices to have unbounded degree.