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

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. One part of the proof proves the result for the bounded case, a result proved independently by Altomare. We give two short proofs of the bounded case of S.B. Rao’s conjecture. Both 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.