The polytope of degree partitions

Abstract

The degree partition of a simple graph is its degree sequence rearranged in weakly decreasing order. The polytope of degree partitions (respectively, degree sequences) is the convex hull of degree partitions (respectively, degree sequences) of all simple graphs on the vertex set [n]. The polytope of degree sequences has been very well studied. In this paper we study the polytope of degree partitions. We show that adding the inequalities x1 ≥ x2 ⋯ ≥ xn to a linear inequality description of the degree sequence polytope yields a linear inequality description of the degree partition polytope and we show that the extreme points of the degree partition polytope are the 2n-1 threshold partitions (these are precisely those extreme points of the degree sequence polytope that have weakly decreasing coordinates). We also show that the degree partition polytope has 2n-2(2n-3) edges and (n2-3n+12)/2 facets, for n ≥ 4. Our main tool is an averaging transformation on real sequences defined by repeatedly averaging over the ascending runs.