Packing of Graphic Sequences

Let π1 and π2 be graphic n-tuples, with π1 = (d (1) 1 , . . . , d (1) n ) and π2 = (d (2) 1 , . . . , d (2) n ) (they need not be monotone). We say that π1 and π2 pack if there exist edge-disjoint graphs G1 and G2 with vertex set {v1, . . . , vn} such that the degrees of vi in G1 and G2 are d (1) i and d (2) i , respectively. We prove that two graphic n-tuples pack if ∆ ≤ √ 2δn− (δ−1), where ∆ and δ denote the largest and smallest entries in π1+π2 (strict inequality when δ = 1); furthermore, the bound is sharp. Kundu proved that a graphic n-tuple π is realizable by a graph having a k-factor if the list obtained by subtracting k from each entry is graphic; we conjecture that when n is even there is a realization containing k edge-disjoint 1-factors. We prove that the conjecture holds when the largest entry is at most n/2 + 1 and also when k ≤ 3.