A semigroup proof of the bounded degree case of S.B. Raoʼs Conjecture on degree sequences and a bipartite analogue

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[6]. In 2008, M. Chudnovsky and P. Seymour proved this long standing Rao’s Conjecture by giving structure theorems for graphic degree sequences[1]. In this paper, we prove and use a fairly simple semigroup lemma 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[7], the first of which implies the bounded degree case of Rao’s Conjecture.