A Semigroup Proof of the Bounded Degree Case of S.B. Rao's Conjecture on Degree Sequences and a Bipartite Analogue

TL;DR

This paper provides a concise, semigroup-theoretic proof of the bounded degree case of Rao's Conjecture on graphic degree sequences, offering an alternative to the existing structure theorem-based proof.

Contribution

It introduces a novel semigroup approach to prove the bounded degree case of Rao's Conjecture, independent of prior structure theorems.

Findings

Proves the bounded degree case of Rao's Conjecture using semigroup theory.

Answers two open questions of N. Robertson, confirming the conjecture in this case.

Abstract

S.B. Rao conjectured in 1971 that graphic degree sequences are well quasi ordered by a relation defined in terms of the induced subgraph relation. In 2008, M. Chudnovsky and P. Seymour proved this long standing Rao's Conjecture by giving structure theorems for graphic degree sequences. In this paper, we prove and use a variant of Dickson's Lemma from commutative semigroup theory to give a short proof of the bounded degree case of Rao's Conjecture that is independent of the Chudnovsky-Seymour structure theory. In fact, we affirmatively answer two questions of N. Robertson, the first of which implies the bounded degree case of Rao's Conjecture.