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 1980 that graphic degree sequences are well quasi ordered by a relation ≼ defined in terms of the induced subgraph relation (Rao, 1981 [7], ). In 2008, M. Chudnovsky and P. Seymour proved this longstanding conjecture by giving structure theorems for graphic degree sequences (Chudnovsky and Seymour, in preparation [2]).In this paper, we prove and use a semigroup lemma to give a short proof of the bounded degree case of Raoʼs Conjecture that is independent of the Chudnovsky–Seymour structure theory. In fact, we affirmatively answer two questions of N. Robertson (2006) [8], the first of which implies the bounded degree case of Raoʼs Conjecture.