Degree sequences of graphs and dominance order

Suppose that the graphical partition H(A) = (a₂¹ ≥ ··· ≥ a_n¹) arises from A = (a₁ ≥ ··· ≥ a_n) by deleting the largest summand a₁ from A and reducing the a₁ largest of the remaining summands by one. Let (a_i+1′ ≥ ··· ≥ a_n′) = H′(A) denote the partition obtained by applying the operator H i times. We prove that the dominance order of partitions is preserved when we switch from A to (a₁ ≥ a₂¹ ≥ ··· ≥ a_i+1′ ≥ ···) =: E(A). This generalizes a recent result by Favaron, Mahéo, and Saclé on the residue of a graph. © 1996 John Wiley & Sons, Inc.